9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions

Verify $\tan \theta \cos \theta = \sin \theta .$

Verify the identity $\csc \theta \cos \theta \tan \theta = 1.$

Verify the following equivalency using the even-odd identities:

Verify the identity $\frac{\sec^{2} \theta - 1}{\sec^{2} \theta} = \sin^{2} \theta$

Show that $\frac{\cot \theta}{\csc \theta} = \cos \theta .$

Create an identity for the expression $2 \tan \theta \sec \theta$ by rewriting strictly in terms of sine.

Verify the identity:

Verify the identity $\frac{\sin^{2} \theta - 1}{\tan \theta \sin \theta - \tan \theta} = \frac{\sin \theta + 1}{\tan \theta} .$

Verify the identity: $\left(1 - \cos^{2} x\right) \left(1 + \cot^{2} x\right) = 1.$

Write the following trigonometric expression as an algebraic expression: $2 \cos^{2} \theta + \cos \theta - 1.$

Rewrite the trigonometric expression using the difference of squares: $4 \cos^{2} \theta - 1.$

Rewrite the trigonometric expression using the difference of squares: $25 - 9 \sin^{2} \theta .$

Simplify the expression by rewriting and using identities:

Use algebraic techniques to verify the identity: $\frac{\cos \theta}{1 + \sin \theta} = \frac{1 - \sin \theta}{\cos \theta} .$

(Hint: Multiply the numerator and denominator on the left side by $1 - \sin \theta . \left.\right)$

We know $g \left(x\right) = \cos x$ is an even function, and $f \left(x\right) = \sin x$ and $h \left(x\right) = \tan x$ are odd functions. What about $G \left(x\right) = \cos^{2} x , F \left(x\right) = \sin^{2} x ,$ and $H \left(x\right) = \tan^{2} x ?$ Are they even, odd, or neither? Why?

After examining the reciprocal identity for $\sec t ,$ explain why the function is undefined at certain points.

Use the fundamental identities to fully simplify the expression.

$\sin x \cos x \sec x$

Use the fundamental identities to fully simplify the expression.

$\tan x \sin x + \sec x \cos^{2} x$

Use the fundamental identities to fully simplify the expression.

$\frac{\cot t + \tan t}{\sec \left(- t\right)}$

Use the fundamental identities to fully simplify the expression.

$− \tan \left(− x\right) \cot \left(− x\right)$

Use the fundamental identities to fully simplify the expression.

$\frac{1 + \tan^{2} \theta}{\csc^{2} \theta} + \sin^{2} \theta + \frac{1}{\sec^{2} \theta}$

Use the fundamental identities to fully simplify the expression.

$\frac{1 - \cos^{2} x}{\tan^{2} x} + 2 \sin^{2} x$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{\sec x + \csc x}{1 + \tan x} ; \sin x$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{1}{\sin x \cos x} - \cot x ; \cot x$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\left(\sec x + \csc x\right) \left(\sin x + \cos x\right) - 2 - \cot x ; \tan x$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\frac{1 - \sin x}{1 + \sin x} - \frac{1 + \sin x}{1 - \sin x} ; \sec x \textrm{ }\text{and}\textrm{ } \tan x$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\sec x ; \cot x$

Simplify the first trigonometric expression by writing the simplified form in terms of the second expression.

$\cot x ; \sin x$

Verify the identity.

$\cos x - \cos^{3} x = \cos x \sin^{2} x$

Verify the identity.

$\frac{1 + \sin^{2} x}{\cos^{2} x} = \frac{1}{\cos^{2} x} + \frac{\sin^{2} x}{\cos^{2} x} = 1 + 2 \tan^{2} x$

Verify the identity.

$\cos^{2} x - \tan^{2} x = 2 - \sin^{2} x - \sec^{2} x$

Prove or disprove the identity.

$\csc^{2} x \left(1 + \sin^{2} x\right) = \cot^{2} x$

Prove or disprove the identity.

$\frac{\tan x}{\sec x} \sin \left(- x\right) = \cos^{2} x$

Prove or disprove the identity.

$\frac{1 + \sin x}{\cos x} = \frac{\cos x}{1 + \sin \left(- x\right)}$

Determine whether the identity is true or false. If false, find an appropriate equivalent expression.

$3 \sin^{2} \theta + 4 \cos^{2} \theta = 3 + \cos^{2} \theta$