91Ó°ÊÓ

Verify the identity. $$\cos ^{2} x-\tan ^{2} x=2-\sin ^{2} x-\sec ^{2} x$$

For the following exercises, use a graph to determine whether are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think \(2 x=x+x . )\) $$ f(\theta)=\cos (2 \theta), g(\theta)=\cos ^{2} \theta-\sin ^{2} \theta $$

For the following exercises, simplify each expression. Do not evaluate. $$ \cos ^{2}(9 x)-\sin ^{2}(9 x) $$

Rewrite the product as a sum or difference. $$2 \sin (5 x) \cos (3 x)$$

Rewrite the sum as a product of two functions. Leave in terms of sine and cosine. $$\sin \left(76^{\circ}\right)+\sin \left(14^{\circ}\right)$$

Prove the identity. $$\sin x+\sin (3 x)=4 \sin x \cos ^{2} x$$

Algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator. $$2 \sin (2 x) \sin (3 x)=\cos x-\cos (5 x)$$

For the following exercises, solve exactly on \([0,2 \pi)\) $$ 2 \sin \theta=-1 $$

A woman is watching a launched rocket currently 11 miles in altitude. If she is standing 4 miles from the launch pad, at what angle is she looking up from horizontal?

For the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree. A 23-foot ladder is positioned next to a house. If the ladder slips at 7 feet from the house when there is not enough traction, what angle should the ladder make with the ground to avoid slipping?