All Solutions
Section 5-5: Trigonometric Identities
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
Remember that for a unit circle, $r=1$
$cos theta=dfrac{x}{r}=x$
$sintheta=dfrac{y}{r}=y$
$tan theta=dfrac{sin theta}{costheta}=dfrac{y}{x}$
The restrictions on $theta$ are its values that could make a denominator equal to zero.
L.S: $cot theta=dfrac{1}{tan theta}=dfrac{1}{y/x}=dfrac{x}{y}$
R.S: $dfrac{costheta}{sin theta}=dfrac{x}{y}$
Restrictions: $sin thetaneq 0$
$$
theta ne left{ {begin{array}{c}
{{0^{text{o}}} + {{360}^{text{o}}}n} \
{{{180}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
L.S.: $tanthetacostheta=dfrac{y}{x}cdot x=y$
R.S.: $sin theta=y$
Restrictions: $cos thetaneq 0$
$$
theta ne left{ {begin{array}{c}
{{90^{text{o}}} + {{360}^{text{o}}}n} \
{{{270}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
L.S.: $csc theta=dfrac{1}{sin theta}=dfrac{1}{y}$
R.S.: $dfrac{1}{sin theta}=dfrac{1}{y}$
Restrictions: $sin thetaneq 0$
$$
theta ne left{ {begin{array}{c}
{{0^{text{o}}} + {{360}^{text{o}}}n} \
{{{180}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
L.S.: $costhetacdot sec theta=xcdot dfrac{1}{x}=1$
R.S.: $1$
Restrictions: $cos thetaneq 0$
$$
theta ne left{ {begin{array}{c}
{{90^{text{o}}} + {{360}^{text{o}}}n} \
{{{270}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
$(1-sin alpha)(1+sinalpha)=1-sin^2 alpha$
From the Pythagorean identity $sin^2theta+cos^2theta=1$
$$
1-sin^2alpha=cos^2alpha
$$
$dfrac{tan alpha}{sin alpha}=dfrac{(sin alpha/cosalpha)}{sin alpha}=dfrac{1}{cos alpha}=sec theta$
$$
cos^2alpha+sin^2alpha=1
$$
$$
cot alpha cdot sin alpha=dfrac{cosalpha}{sinalpha}cdot sinalpha=cosalpha
$$
b) $sectheta$
c) $1$
d) $cosalpha$
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
$1-cos^2theta$
$=(1)^2-(costheta)^2$
$$
=(1-costheta)(1+costheta)
$$
$sin^2theta-cos^2theta$
$=(sin theta)^2-(costheta)^2$
$$
=(sintheta+costheta)(sintheta-costheta)
$$
$sin^2theta-2sintheta+1$
$=(sintheta)^2-2(sintheta)(1)+(1)^2$
$$
=(sintheta-1)^2
$$
$costheta-cos^2theta$
$$
=costheta(1-costheta)
$$
b) $(sintheta+costheta)(sintheta-costheta)$
c) $(sintheta-1)^2$
d) $costheta(1-costheta)$
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
L.S: $dfrac{cos^2phi}{1-sinphi}=dfrac{1-sin^2phi}{1-sinphi}$
Factor using $a^2-b^2=(a+b)(a-b)$
$implies dfrac{1-sin^2phi}{1-sinphi}$=$dfrac{(1-sinphi)(1+sinphi)}{1-sinphi}$
$=1+sinphi=$ R.S.
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
Remember that for a unit circle, $r=1$
$cos theta=dfrac{x}{r}=x$
$sintheta=dfrac{y}{r}=y$
$tan theta=dfrac{sin theta}{costheta}=dfrac{y}{x}$
The restrictions on $theta$ are its values that could make any denominator equal to zero.
L.S. = $dfrac{sin x}{tan x}=dfrac{sin x}{sin x / cos x}=cos x$ = R.S.
Restrictions: $cos x neq 0$ and $sin x neq 0$
$$
x neq left{ {begin{array}{c}
{{90^{text{o}}} + {{360}^{text{o}}}n} \
{{{270}^{text{o}}} + {{360}^{text{o}}}n} \
{{0^{text{o}}} + {{360}^{text{o}}}n} \
{{{180}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
L.S. = $dfrac{tan theta}{cos theta}=dfrac{sin theta/costheta}{cos theta}=dfrac{sin theta}{cos^2theta}$
Use $sin^2theta+cos^2theta=1$
$dfrac{sin theta}{cos^2theta}=dfrac{sintheta}{1-sin^2theta}$ = R.S.
Restrictions: $cos thetaneq 0$
$$
theta ne left{ {begin{array}{c}
{{90^{text{o}}} + {{360}^{text{o}}}n} \
{{{270}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
R.S. = $dfrac{1+sin alpha}{cosalpha}=dfrac{1}{cosalpha}+dfrac{sinalpha}{cosalpha}$
Use $tan theta=dfrac{sintheta}{costheta}$
$dfrac{1}{cosalpha}+dfrac{sinalpha}{cosalpha}=dfrac{1}{cosalpha}+tan alpha =$ L.S.
Restrictions: $cosalpha neq 0$
$$
alpha ne left{ {begin{array}{c}
{{90^{text{o}}} + {{360}^{text{o}}}n} \
{{{270}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
Use $tan theta=dfrac{sintheta}{costheta}$
R.S. = $sintheta costheta tan theta=sintheta costheta cdot dfrac{sin theta}{costheta}= sin^2 theta$
Use $sin^2theta+cos^2theta=1$
$sin^2theta=1-cos^2theta =$ L.S.
Restrictions: $cos xneq 0$
$$
x neq left{ {begin{array}{c}
{{90^{text{o}}} + {{360}^{text{o}}}n} \
{{{270}^{text{o}}} + {{360}^{text{o}}}n}
end{array}} right.
$$
where $n$ is an integer
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
$sin theta cot theta – sin theta cos theta$
$color{#c34632}{ text{Use};; cottheta=dfrac{costheta}{sintheta}}$
$=sin theta cdot dfrac{costheta}{sintheta}-sinthetacostheta$
$=costheta-sinthetacostheta$
$$
=costheta(1-sintheta)
$$
$costheta(1+sectheta)(costheta-1)$
$=costhetaleft(1+dfrac{1}{costheta}right)(costheta-1)$
$=(costheta+1)(costheta-1)$
$color{#c34632}{text{Use} ;; (a+b)(a-b)=a^2-b^2}$
$=cos^2theta-1$
$color{#c34632}{text{Use};; sin^2theta+cos^2theta=1}$
$$
=-sin^2theta
$$
$color{#c34632}{text{Use};; (a+b)(a-b)=a^2-b^2}$
$(sin x+cos x)(sin x-cos x)+2cos^2 x$
$=sin^2 x -cos^2 x+2cos^2 x$
$=sin^2 x+cos^2x$
$color{#c34632}{text{Use};; sin^2theta+cos^2theta=1}$
$$
=1
$$
$color{#c34632}{text{Let } x=csctheta}$
$dfrac{csc^2theta-3csctheta+2}{csc^2theta-1}=dfrac{x^2-3x+2}{x^2-1}$
$color{#c34632}{text{Factor the numerator and denominator}}$
$=dfrac{(x-1)(x-2)}{(x-1)(x+1)}=dfrac{x-2}{x+1}$
$color{#c34632}text{Revert back to } x=csctheta$
$$
=dfrac{csctheta-2}{csctheta+1}
$$
b) $-sin^2theta$
c) 1
d) $dfrac{csctheta-2}{csctheta+1}$
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
$color{#c34632} text{Use } sin^2theta+cos^2theta=1$
L.S. = $dfrac{sin^2phi}{1-cosphi}=dfrac{1-cos^2phi}{1-cosphi}$
$color{#c34632}text{Factor using } a^2-b^2=(a+b)(a-b)$
$=dfrac{(1+cosphi)(1-cos phi)}{1-cos phi}=1+cos phi$ = R.S.
$color{#c34632}text{Use } tan^2theta+1=sec^2theta$
L.S. = $dfrac{tan^2alpha}{1+tan^2alpha}=dfrac{tan^2alpha}{sec^2alpha}=left(dfrac{tan alpha}{sec alpha}right)^2$
$color{#c34632}text{Use } tan theta=dfrac{sin theta}{costheta} ;text{and};sectheta=dfrac{1}{costheta}$
$=left(dfrac{sin alpha / cos alpha}{1/cosalpha}right)^2=sin^2alpha$ = R.S.
$color{#c34632};text{Use } (a-b)(a+b)=a^2-b^2;text{and}; sin^2theta+cos^2theta=1$
R.S. = $(1-sin x)(1+sin x)=1-sin^2 x =cos^2 x$ = L.S.
$sin^2theta+2cos^2theta-1$
$=(1-cos^2theta)+2cos^2theta-1$
$=1+cos^2theta-1$
$=cos^2theta$ = R.S.
L.S. = $sin^4alpha-cos^4alpha$
$=(sin^2alpha+cos^2alpha)(sin^2alpha-cos^2alpha)$
$color{#c34632};text{Use}; sin^2theta+cos^2theta=1$
$=(1)(sin^2alpha-cos^2alpha)=$ R.S.
$color{#c34632}text{Use the quotient identities for tangent}$
$=dfrac{sin theta}{costheta}+dfrac{costheta}{sin theta}$
$color{#c34632}text{Use} sin^2theta+cos^2theta=1$
$=dfrac{sin^2theta+cos^2theta}{sintheta costheta}$
$=dfrac{1}{sinthetacostheta}$ = R.S.
b) This approach can only be used if there are no trigonometric ratios in the denominator.
b) This approach can only be used if there are no trigonometric ratios in the denominator.
For $theta=45^circ$, both sides are defined.
L.S. $csc^2 45^circ+sec^2 45^circ =(sqrt{2})^2+(sqrt{2})^2=4$
R.S. $1$
L.S. $neq$ R.S.
Therefore, it is NOT an identity.
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
$=sin^2 xleft(dfrac{tan^2x+1}{tan^2x}right)$
$color{#c34632}text{Use }1+tan^2theta=sec^2theta;text{and};sectheta=dfrac{1}{costheta}$
$=sin^2 x left(dfrac{sec^2 x}{tan^2x}right)$
$=sin^2 x left(dfrac{sec x}{tan x}right)^2$
$=sin^2x left(dfrac{1/cos x}{sin x/cos x}right)^2$
$=sin^2xcdot dfrac{1}{sin^2x}$
$=1=$ R.S.
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
$color{#c34632};text{Use};sin^2theta+cos^2theta=1$
L.S. = $dfrac{sin^2theta+2costheta-1}{sin^2theta+3costheta-3}=dfrac{(1-cos^2theta)+2costheta-1}{(1-cos^2theta)+3costheta-3}$
$=dfrac{-cos^2theta+2costheta}{-cos^2theta+3costheta-2}$
$color{#c34632};text{Let};x=costheta$
$=dfrac{-x^2+2x}{-x^2+3x-2}=dfrac{(-1)(x^2-2x)}{(-1)(x^2-3x+2)}$
$=dfrac{x(x-2)}{(x-1)(x-2)}=dfrac{x}{x-1}=dfrac{costheta}{costheta-1}$
R.S. = $dfrac{cos^2theta+costheta}{-sin^2theta}$
$color{#c34632}text{Use }sin^2theta+cos^2theta=1$
$=dfrac{cos^2theta+costheta}{-(1-cos^2theta)}$
$=dfrac{cos^2theta+costheta}{cos^2theta-1}$
$=dfrac{costheta(costheta+1)}{(costheta+1)(costheta-1)}$
$=dfrac{costheta}{costheta-1}$
$costheta neq 1$ and $sin thetaneq 0$
$color{#c34632}text{Let};y=sin alpha$
$=dfrac{2y^2-2y^4-1}{1-y^2}$
$=dfrac{2y^2(1-y^2)-1}{1-y^2}$
$=dfrac{2y^2(1-y^2)}{1-y^2}-dfrac{1}{1-y^2}$
$=2y^2-dfrac{1}{1-y^2}$
$=2sin^2theta-dfrac{1}{1-sin^2theta}$
$color{#c34632}text{Use};sin^2theta+cos^2theta=1$
$=2sin^2theta-dfrac{1}{cos^2theta}$
$color{#c34632}text{Use};sectheta=dfrac{1}{costheta}$
$=2sin^2theta-sec^2theta$
$color{#c34632}text{Use};tan^2theta+1=sec^2theta$
$=2sin^2theta-(tan^2theta+1)$
$color{#c34632}text{Use};sin^2theta+cos^2theta=1$
$=2sin^2theta-tan^2theta+(sin^2theta+cos^2theta)$
$=sin^2theta+cos^2theta-tan^2theta$ = L.S.
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
i) Multiply both sides by $dfrac{1}{sintheta}$
$left(dfrac{1}{sintheta}right) (sin^2theta+cos^2theta)=dfrac{1}{sintheta}$
$$
sintheta+csctheta cos^2theta=csctheta
$$
$left(dfrac{1}{cos^2theta}right) (sin^2theta+cos^2theta)=dfrac{1}{cos^2theta}$
$tan^2theta+1=dfrac{1}{cos^2theta}$
Subtract both sides by $1$
$tan^2theta=dfrac{1}{cos^2theta}-1$
$left(secthetaright) (sin^2theta+cos^2theta)=1cdot sec theta$
$sin^2thetasectheta+costheta=sectheta$
defarraystretch{2.5}%
begin{tabular}{|l|l|}
hline
$bold{Reciprocal; Identities}$ & $sec theta=dfrac{1}{costheta}$ \ cline{2-2}
& $csc theta=dfrac{1}{sin theta}$ \ cline{2-2}
& $cot theta=dfrac{1}{tan theta}$ \ hline
$bold{Quotient; Identities}$ & $tan theta=dfrac{sintheta}{costheta}$ \ cline{2-2}
& $cottheta=dfrac{costheta}{sin theta}$ \ hline
$bold{Pythagorean; Identities}$ & $sin^2theta+cos^2theta=1$ \ cline{2-2}
& $1+tan^2theta=sec^2theta$ \ cline{2-2}
& $1+cot^2theta=csc^2theta$ \ hline
end{tabular}
end{table}
{text{i)}} hfill \
hfill \
{text{L}}{text{.S}}{text{.}} = left( {1 – {{cos }^2}x} right)left( {1 – {{tan }^2}x} right){text{ }} hfill \
{text{ }} hfill \
{color{red}text{[Use }}color{red}{color{red}sin ^2}x + {cos ^2}x = 1 Rightarrow 1 – {cos ^2}x = {sin ^2}x] hfill \
hfill \
= {sin ^2}xleft( {1 – {{tan }^2}x} right){text{ }} hfill \
hfill \
color{red} left[color{red} {{text{Use }}color{red}tan x = frac{{sin x }}{{cos x }}} right] hfill \
{text{ }} hfill \
= {sin ^2}xleft( {1 – frac{{{{sin }^2}x}}{{{{cos }^2}x}}} right){text{ }} hfill \
{text{ }} hfill \
= {sin ^2}xleft( {frac{{{{cos }^2}x – {{sin }^2}x}}{{{{cos }^2}x}}} right) hfill \
hfill \
= frac{{{{sin }^2}x{{cos }^2}x – {{sin }^4}x}}{{{{cos }^2}x}} hfill \
hfill \
= frac{{{{sin }^2}x{{cos }^2}x – {{sin }^4}x}}{{{{cos }^2}x}} hfill \
hfill \
{text{color{red}[Use }}color{red}{sin ^2}x + {cos ^2}x = 1 Rightarrow {cos ^2}x = 1 – {sin ^2}x] hfill \
hfill \
= frac{{{{sin }^2}xleft( {1 – {{sin }^2}x} right) – {{sin }^4}x}}{{1 – {{sin }^2}x}} hfill \
{text{ }} hfill \
= frac{{{{sin }^2}x – {{sin }^4}x – {{sin }^4}x}}{{1 – {{sin }^2}x}} hfill \
hfill \
= frac{{{{sin }^2}x – 2{{sin }^4}x}}{{1 – {{sin }^2}x}} = {text{R}}{text{.S}}{text{.}} hfill \
hfill \
end{gathered} ]
{text{ii)}} hfill \
1 – 2{cos ^2}phi = {sin ^4}phi – {cos ^4}phi hfill \
hfill \
{text{R}}{text{.S}}{text{. }} = {sin ^4}phi – {cos ^4}phi hfill \
hfill \
color{red} left[ {{text{Factor using }}{a^4} – {b^4} = left( {{a^2} + {b^2}} right)left( {{a^2} – {b^2}} right)} right] hfill \
hfill \
= left( {{{sin }^2}phi + {{cos }^2}phi } right)left( {{{sin }^2}phi – {{cos }^2}phi } right) hfill \
hfill \
color{red} left[ {{text{Use }}{{sin }^2}theta + {{cos }^2}theta = 1} right] hfill \
hfill \
= left( 1 right)left( {{{sin }^2}theta – {{cos }^2}theta } right) hfill \
hfill \
color{red} left[ {{text{Use }}{{sin }^2}theta + {{cos }^2}theta = 1 Rightarrow {{sin }^2}theta = 1 – {{cos }^2}theta } right] hfill \
hfill \
= left( {1 – {{cos }^2}phi } right) – {cos ^2}phi hfill \
hfill \
= 1 – 2{cos ^2}phi = {text{ L}}{text{.S}}{text{.}} hfill \
end{gathered} ]
{text{iii)}} hfill \
hfill \
frac{{sin theta tan theta }}{{sin theta + tan theta }} = sin theta tan theta hfill \
hfill \
{text{if }}theta = {text{4}}{{text{5}}^{text{o}}} hfill \
sin {45^{text{o}}} = frac{{sqrt 2 }}{2} hfill \
tan theta = 1 hfill \
hfill \
{text{L}}{text{.S}}{text{.}} = frac{{left( {frac{{sqrt 2 }}{2}} right)left( 1 right)}}{{frac{{sqrt 2 }}{2} + 1}} = frac{{sqrt 2 /2}}{{left( {sqrt 2 + 2} right)/2}} = frac{{sqrt 2 }}{{2 + sqrt 2 }} = 0.414 hfill \
{text{R}}{text{.S}}{text{.}} = left( {frac{{sqrt 2 }}{2}} right)left( 1 right) = frac{{sqrt 2 }}{2} = 0.707 hfill \
hfill \
{text{Thus, it is NOT an identity}}{text{.}} hfill \
end{gathered} ]
{text{iv)}} hfill \
hfill \
{text{L}}{text{.S}}{text{.}} = frac{{1 + 2sin beta cos beta }}{{sin beta + cos beta }} hfill \
hfill \
color{red}left[ {{text{Use }}1 = {{sin }^2}beta + {{cos }^2}beta } right] hfill \
hfill \
= frac{{{{sin }^2}beta + {{cos }^2}beta + 2sin beta cos beta }}{{sin beta + cos beta }} hfill \
hfill \
color{red}left[ {{text{Factor using }}left( {{a^2} + 2ab + {b^2}} right) = {{left( {a + b} right)}^2}} right] hfill \
hfill \
= frac{{{{left( {sin beta + cos beta } right)}^2}}}{{sin beta + cos beta }} hfill \
hfill \
= sin beta + cos beta = {text{ R}}{text{.S}}{text{.}} hfill \
end{gathered} ]
{text{v)}} hfill \
hfill \
{text{L}}{text{.S}}{text{.}} = frac{{1 – cos beta }}{{sin beta }} hfill \
hfill \
color{red}left[ {{text{Multiply by }}frac{{1 + cos beta }}{{1 + cos beta }}{text{ }}} right] hfill \
hfill \
= frac{{1 – cos beta }}{{sin beta }} cdot frac{{1 + cos beta }}{{1 + cos beta }} hfill \
hfill \
= frac{{1 – {{cos }^2}beta }}{{sin beta left( {1 + cos beta } right)}} hfill \
hfill \
color{red}left[ {{text{Use }}{{sin }^2}beta = 1 – {{cos }^2}beta {text{ }}} right] hfill \
hfill \
= frac{{{{sin }^2}beta }}{{sin beta left( {1 + cos beta } right)}} hfill \
hfill \
= frac{{sin beta }}{{1 + cos beta }} = {text{R}}{text{.S}}{text{.}} hfill \
end{gathered} ]
{text{vi)}} hfill \
hfill \
color{red}left[ {{text{Multiply by }}frac{{1 – cos x}}{{1 – cos x}}} right] hfill \
hfill \
{text{L}}{text{.S}}{text{.}} = frac{{sin x}}{{1 + cos x}} cdot frac{{1 – cos x}}{{1 – cos x}} hfill \
hfill \
= frac{{sin xleft( {1 – cos x} right)}}{{1 – {{cos }^2}x}} hfill \
hfill \
color{red} left[ {{text{Use }}1 – {{cos }^2}x = {{sin }^2}x} right] hfill \
hfill \
= frac{{sin xleft( {1 – cos x} right)}}{{{{sin }^2}x}} hfill \
hfill \
= frac{{1 – cos x}}{{sin x}} hfill \
hfill \
color{red}left[ {{text{Use }}frac{1}{{sin x}} = csc x{text{ and }}frac{{cos x}}{{sin x}} = cot x} right] hfill \
hfill \
= csc x – cot x = {text{ R}}{text{.S}}{text{.}} hfill \
end{gathered} ]
b) The restrictions are those that could make the denominator zero:
i) $1-sin^2xneq 0implies$ $sin^2xneq 1$
ii) no restrictions, i.e. true for any real number
iii) not applicable
iv) $sin theta+cos thetaneq 0implies sinthetaneq -costheta$
v) $sin betaneq 0$ and $1+costhetaneq 0implies costhetaneq -1$
vi) $1+cos xneq 0implies cos xneq -1$