The rate of change of the function$f(x)=\sin x+\csc x$ with respect to change in the variable $x$ is given by the expression $\cos x-\csc x\cot x.$ Show that the expression for the rate of change can also be $-\cos x\cot^2x.$

rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.

$$\newline\frac{3}{\sec x-\tan x}$$

Write the trigonometric expression as an algebraic expression.

$$\newline\cos(\arccos x-\arctan x)$$

Find the exact values of $\sin 2 u, \cos 2 u$, and $\tan 2 u$ using the double-angle formulas.

$$\cot u=2, \quad \pi<u<\frac{3 \pi}{2}$$

Use the cofunction identities to evaluate the expression without using a calculator.

$$\tan ^2 63^{\circ}+\cot ^2 16^{\circ}-\sec ^2 74^{\circ}-\csc ^2 27^{\circ}$$

Use double-angle formulas to verify the identity algebraically and use a graphing utility to confirm your result graphically.

$$\sin 4 x=8 \cos ^3 x \sin x-4 \cos x \sin x$$

Verify the identity.

$$\sin \theta \sec \theta=\tan \theta$$

rewrite the expression so that it is not in fractional form. There is more than one correct form of each answer.

$$\newline\frac{5}{\tan x+\sec x}$$

Fill in the blank to complete the trigonometric formula.

$$\cos u \cos v=$$

Use the half-angle formulas to simplify the expression.

$$-\sqrt{\frac{1-\cos (x-1)}{2}}$$

Use the given values and trigonometric identities to evaluate (if possible) all six trigonometric functions.

$$\sin x=\frac{5}{13}, \quad \cos x=\frac{12}{13}$$

Fill in the blank to complete the trigonometric identity.

$$\sin \left(\frac{\pi}{2}-u\right)=\underline{\qquad}$$

Use the cofunction identities to evaluate the expression without using a calculator,

$$\newline\cos^220^\circ+\cos^252^\circ+\cos^238^\circ+\cos^270^\circ$$

Fill in the blank to complete the trigonometric identity.

$$\csc (-u)=\underline{\qquad}$$