Section 1
Verifying Trigonometric Identities
Verify the Identity.$$\csc \theta-\sin \theta=\cot \theta \cos \theta$$
Verify the Identity.$$\sin x+\cos x \cot x=\csc x$$
Verify the Identity.$$\frac{\sec ^{2} 2 u-1}{\sec ^{2} 2 u}=\sin ^{2} 2 u$$
Verify the Identity.$$\tan t+2 \cos t \csc t=\sec t \csc t+\cot t$$
Verify the Identity.$$\frac{\csc ^{2} \theta}{1+\tan ^{2} \theta}=\cot ^{2} \theta$$
Verify the Identity.$$(\tan u+\cot u)(\cos u+\sin u)=\csc u+\sec u$$
Verify the Identity.$$\frac{1+\cos 3 t}{\sin 3 t}+\frac{\sin 3 t}{1+\cos 3 t}=2 \csc 3 t$$
Verify the Identity.$$\tan ^{2} \alpha-\sin ^{2} \alpha=\tan ^{2} \alpha \sin ^{2} \alpha$$
Verify the Identity.$$\frac{1}{1-\cos \gamma}+\frac{1}{1+\cos \gamma}=2 \csc ^{2} \gamma$$
Verify the Identity.$$\frac{1+\csc 3 \beta}{\sec 3 \beta}-\cot 3 \beta=\cos 3 \beta$$
Verify the Identity.$$(\sec u-\tan u)(\csc u+1)=\cot u$$
Verify the Identity.$$\frac{\cot \theta-\tan \theta}{\sin \theta+\cos \theta}=\csc \theta-\sec \theta$$
Verify the Identity.$$\csc ^{4} t-\cot ^{4} t=\csc ^{2} t+\cot ^{2} t$$
Verify the Identity.$$\cos ^{4} 2 \theta+\sin ^{2} 2 \theta=\cos ^{2} 2 \theta+\sin ^{4} 2 \theta$$
Verify the Identity.$$\frac{\cos \beta}{1-\sin \beta}=\sec \beta+\tan \beta$$
Verify the Identity.$$\frac{1}{\csc y-\cot y}=\csc y+\cot y$$
Verify the Identity.$$\frac{\tan ^{2} x}{\sec x+1}=\frac{1-\cos x}{\cos x}$$
Verify the Identity.$$\frac{\cot x}{\csc x+1}=\frac{\csc x-1}{\cot x}$$
Verify the Identity.$$\frac{\cot 4 u-1}{\cot 4 u+1}=\frac{1-\tan 4 u}{1+\tan 4 u}$$
Verify the Identity.$$\frac{1+\sec 4 x}{\sin 4 x+\tan 4 x}=\csc 4 x$$
Verify the Identity.$$\sin ^{4} r-\cos ^{4} r=\sin ^{2} r-\cos ^{2} r$$
Verify the Identity.$$\sin ^{4} \theta+2 \sin ^{2} \theta \cos ^{2} \theta+\cos ^{4} \theta=1$$
Verify the Identity.$$\tan ^{4} k-\sec ^{4} k=1-2 \sec ^{2} k$$
Verify the Identity.$$\sec ^{4} u-\sec ^{2} u=\tan ^{2} u+\tan ^{4} u$$
Verify the Identity.$$(\sec t+\tan t)^{2}=\frac{1+\sin t}{1-\sin t}$$
Verify the Identity.$$\sec ^{2} \gamma+\tan ^{2} \gamma=\left(1-\sin ^{4} \gamma\right) \sec ^{4} \gamma$$
Verify the Identity.$$\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{3}=1$$
Verify the Identity.$$\frac{\sin t}{1-\cos t}=\csc t+\cot t$$
Verify the Identity.$$\frac{1+\csc \beta}{\cot \beta+\cos \beta}=\sec \beta$$
Verify the Identity.$$\frac{\cos ^{3} x-\sin ^{3} x}{\cos x-\sin x}=1+\sin x \cos x$$
Verify the Identity.$$(\csc t-\cot t)^{4}(\csc t+\cot t)^{4}=1$$
Verify the Identity.$$(a \cos t-b \sin t)^{2}+(a \sin t+b \cos t)^{2}=a^{2}+b^{2}$$
Verify the Identity.$$\frac{\sin \alpha \cos \beta+\cos \alpha \sin \beta}{\cos \alpha \cos \beta-\sin \alpha \sin \beta}=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}$$
Verify the Identity.$$\frac{\tan u-\tan v}{1+\tan u \tan v}=\frac{\cot v-\cot u}{\cot u \cot v+1}$$
Verify the Identity.$$\frac{\tan \alpha}{1+\sec \alpha}+\frac{1+\sec \alpha}{\tan \alpha}=2 \csc \alpha$$
Verify the Identity.$$\frac{\csc x}{1+\csc x}-\frac{\csc x}{1-\csc x}=2 \sec ^{2} x$$
Verify the Identity.$$\frac{1}{\tan \beta+\cot \beta}=\sin \beta \cos \beta $$
Verify the Identity.$$\frac{\cot y-\tan y}{\sin y \cos y}=\csc ^{2} y-\sec ^{2} y$$
Verify the Identity.$$\sec \theta+\csc \theta-\cos \theta-\sin \theta=\sin \theta \tan \theta+\cos \theta \cot \theta$$
Verify the Identity.$$\sin ^{3} t+\cos ^{3} t=(1-\sin t \cos t)(\sin t+\cos t)$$
Verify the Identity.$$\left(1-\tan ^{2} \phi\right)^{2}=\sec ^{4} \phi-4 \tan ^{2} \phi$$
Verify the Identity.$$\cos ^{4} w+1-\sin ^{4} w=2 \cos ^{2} w$$
Verify the Identity.$$\frac{\cot (-t)+\tan (-t)}{\cot t}=-\sec ^{2} t$$
Verify the Identity.$$\frac{\csc (-t)-\sin (-t)}{\sin (-t)}=\cot ^{2} t$$
Verify the Identity.$$\log 10^{\operatorname{tan} t}=\tan t$$
Verify the Identity.$$10^{\log |\sin t|}=|\sin t|$$
Verify the Identity.$$\ln \cot x=-\ln \tan x$$
Verify the Identity.$$\text { In } \sec \theta=-\ln \cos \theta$$
Verify the Identity.$$\ln |\sec \theta+\tan \theta|=-\ln |\sec \theta-\tan \theta|$$
Verify the Identity.$$\ln |\csc x-\cot x|=-\ln |\csc x+\cot x|$$
Show that the equation is not an Identity.$$\cos t=\sqrt{1-\sin ^{2} t}$$
Show that the equation is not an Identity.$$\sqrt{\sin ^{2} t+\cos ^{2} t}=\sin t+\cos t$$
Show that the equation is not an Identity.$$\sqrt{\sin ^{2} t}=\sin t$$
Show that the equation is not an Identity.$$\sec t=\sqrt{\tan ^{2} t+1}$$
Show that the equation is not an Identity.$$(\sin \theta+\cos \theta)^{2}=\sin ^{2} \theta+\cos ^{2} \theta$$
Show that the equation is not an Identity.$$\log \left(\frac{1}{\sin t}\right)=\frac{1}{\log \sin t}$$
Show that the equation is not an Identity.$$\cos (-t)=-\cos t$$
Show that the equation is not an Identity.$$\sin (t+\pi)=\sin t$$
Show that the equation is not an Identity.$$\cos (\sec t)=1$$
Show that the equation is not an Identity.$$\cot (\tan \theta)=1$$
Either show that the equation $i s$ an identity or show that the equation $is\quad not$ an identity.$$(\sec x+\tan x)^{2}=2 \tan x(\tan x+\sec x)$$
Either show that the equation $i s$ an identity or show that the equation $is\quad not$ an identity.$$\frac{\tan ^{2} x}{\sec x-1}=\sec x$$
Either show that the equation $i s$ an identity or show that the equation $is\quad not$ an identity.$$\cos x(\tan x+\cot x)=\csc x$$
Either show that the equation $i s$ an identity or show that the equation $is\quad not$ an identity.$$\csc ^{2} x+\sec ^{2} x=\csc ^{2} x \sec ^{2} x$$
Make the trigonometric substitution $x=a \sin \theta$ for $-\pi / 2<\theta<\pi / 2$ and $a>0 .$ Use fundamental identities to simplify the resulting expression.$$\left(a^{2}-x^{2}\right)^{3 / 2}$$
Make the trigonometric substitution $x=a \sin \theta$ for $-\pi / 2<\theta<\pi / 2$ and $a>0 .$ Use fundamental identities to simplify the resulting expression.$$\frac{\sqrt{a^{2}-x^{2}}}{x}$$
Make the trigonometric substitution $x=a \sin \theta$ for $-\pi / 2<\theta<\pi / 2$ and $a>0 .$ Use fundamental identities to simplify the resulting expression.$$\frac{x^{2}}{\sqrt{a^{2}-x^{2}}}$$
Make the trigonometric substitution $x=a \sin \theta$ for $-\pi / 2<\theta<\pi / 2$ and $a>0 .$ Use fundamental identities to simplify the resulting expression.$$\frac{1}{x \sqrt{a^{2}-x^{2}}}$$
Make the trigonometric substitution $$x=a \tan \theta \quad \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\sqrt{a^{2}+x^{2}}$$
Make the trigonometric substitution $$x=a \tan \theta \quad \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\frac{1}{\sqrt{a^{2}+x^{2}}}$$
Make the trigonometric substitution $$x=a \tan \theta \quad \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\frac{1}{x^{2}+a^{2}}$$
Make the trigonometric substitution $$x=a \tan \theta \quad \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\frac{\left(x^{2}+a^{2}\right)^{3 / 2}}{x}$$
Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\sqrt{x^{2}-a^{2}}$$
Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\frac{1}{x^{2} \sqrt{x^{2}-a^{2}}}$$
Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$x^{3} \sqrt{x^{2}-a^{2}}$$
Make the trigonometric substitution $$x=a \sec \theta \quad \text { for } 0<\theta<\pi / 2 \text { and } a>0.$$ Simplify the resulting expression.$$\frac{\sqrt{x^{2}-a^{2}}}{x^{2}}$$
Use the graph of $f$ to find the simplest expression $g(x)$ such that the equation $f(x)=g(x)$ is an Identity. Verify this identity.$$f(x)=\frac{\sin ^{2} x-\sin ^{4} x}{\left(1-\sec ^{2} x\right) \cos ^{4} x}$$
Use the graph of $f$ to find the simplest expression $g(x)$ such that the equation $f(x)=g(x)$ is an Identity. Verify this identity.$$f(x)=\frac{\sin x-\sin ^{3} x}{\cos ^{4} x+\cos ^{2} x \sin ^{2} x}$$
Use the graph of $f$ to find the simplest expression $g(x)$ such that the equation $f(x)=g(x)$ is an Identity. Verify this identity.$$f(x)=\sec x\left(\sin x \cos x+\cos ^{2} x\right)-\sin x$$
Use the graph of $f$ to find the simplest expression $g(x)$ such that the equation $f(x)=g(x)$ is an Identity. Verify this identity.$$f(x)=\frac{\sin ^{3} x+\sin x \cos ^{2} x}{\csc x}+\frac{\cos ^{3} x+\cos x \sin ^{2} x}{\sec x}$$