Chapter 7

For the following exercises, simplify each expression. Do not evaluate. $$ \cos ^{2}\left(28^{\circ}\right)-\sin ^{2}\left(28^{\circ}\right) $$

For the following exercises, simplify each expression. Do not evaluate. $$ 2 \cos ^{2}\left(37^{\circ}\right)-1 $$

For the following exercises, simplify each expression. Do not evaluate. $$ 1-2 \sin ^{2}\left(17^{\circ}\right) $$

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

For the following exercises, simplify each expression. Do not evaluate. $$ 4 \sin (8 x) \cos (8 x) $$

For the following exercises, simplify each expression. Do not evaluate. $$ 6 \sin (5 x) \cos (5 x) $$

For the following exercises, prove the identity given. $$ (\sin t-\cos t)^{2}=1-\sin (2 t) $$

For the following exercises, prove the identity given. $$ \sin (2 x)=-2 \sin (-x) \cos (-x) $$

For the following exercises, prove the identity given. $$ \cot x-\tan x=2 \cot (2 x) $$

For the following exercises, prove the identity given. $$ \frac{\sin (2 \theta)}{1+\cos (2 \theta)} \tan ^{2} \theta=\tan \theta $$

For the following exercises, rewrite the expression with an exponent no higher than 1. $$ \cos ^{2}(5 x) $$

For the following exercises, rewrite the expression with an exponent no higher than 1. $$ \cos ^{2}(6 x) $$

For the following exercises, rewrite the expression with an exponent no higher than 1. $$ \sin ^{4}(3 x) $$

For the following exercises, rewrite the expression with an exponent no higher than 1. $$ \cos ^{2} x \sin ^{4} x $$

For the following exercises, rewrite the expression with an exponent no higher than 1. $$ \cos ^{4} x \sin ^{2} x $$

For the following exercises, rewrite the expression with an exponent no higher than 1. $$ \tan ^{2} x \sin ^{2} x $$

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. $$ \tan ^{4} x \cos ^{2} x $$

For the following exercises, reduce the equations to powers of one, and then check the answer graphically. $$ \cos ^{2} x \sin (2 x) $$

For the following exercises, algebraically find an equivalent function, only in terms of \(\sin x\) and or \(\cos x\) , and then check the answer by graphing both equations. $$ \sin (4 x) $$

For the following exercises, algebraically find an equivalent function, only in terms of \(\sin x\) and or \(\cos x\) , and then check the answer by graphing both equations. $$ \cos (4 x) $$

For the following exercises, prove the identities. $$ \cos (2 \alpha)=\frac{1-\tan ^{2} \alpha}{1+\tan ^{2} \alpha} $$

For the following exercises, prove the identities. $$ \tan (2 x)=\frac{2 \sin x \cos x}{2 \cos ^{2} x-1} $$

For the following exercises, prove the identities. $$ \left(\sin ^{2} x-1\right)^{2}=\cos (2 x)+\sin ^{4} x $$

For the following exercises, prove the identities. $$ \sin (3 x)=3 \sin x \cos ^{2} x-\sin ^{3} x $$

For the following exercises, prove the identities. $$ \frac{1+\cos (2 t)}{\sin (2 t)-\cos t}=\frac{2 \cos t}{2 \sin t-1} $$

For the following exercises, prove the identities. $$ \sin (16 x)=16 \sin x \cos x \cos (2 x) \cos (4 x) \cos (8 x) $$

For the following exercises, prove the identities. $$ \cos (16 x)=\left(\cos ^{2}(4 x)-\sin ^{2}(4 x)-\sin (8 x)\right)\left(\cos ^{2}(4 x)-\sin ^{2}(4 x)+\sin (8 x)\right) $$

Starting with the product to sum formula \(\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]\) , explain how to determine the formula for \(\cos \alpha \sin \beta .\)

Explain two different methods of calculating \(\cos \left(195^{\circ}\right) \cos \left(105^{\circ}\right),\) one of which uses the product to sum. Which method is easier?

Explain a situation where we would convert an equation from a product to a sum, and give an example.

Rewrite the product as a sum or difference. $$16 \sin (16 x) \sin (11 x)$$

Rewrite the product as a sum or difference. $$20 \cos (36 t) \cos (6 t)$$

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

Rewrite the product as a sum or difference. $$10 \cos (5 x) \sin (10 x)$$

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

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

Rewrite the sum or difference as a product. $$\cos (6 t)+\cos (4 t)$$

Rewrite the sum or difference as a product. $$\sin (3 x)+\sin (7 x)$$

Rewrite the sum or difference as a product. $$\cos (7 x)+\cos (-7 x)$$

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

Rewrite the sum or difference as a product. $$\cos (3 x)+\cos (9 x)$$

Rewrite the sum or difference as a product. $$\sin h-\sin (3 h)$$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly. $$\cos \left(45^{\circ}\right) \cos \left(15^{\circ}\right)$$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly. $$\cos \left(45^{\circ}\right) \sin \left(15^{\circ}\right)$$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly. $$\sin \left(-345^{\circ}\right) \sin \left(-15^{\circ}\right)$$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly. $$\sin \left(195^{\circ}\right) \cos \left(15^{\circ}\right)$$

Evaluate the product for the following using a sum or difference of two functions. Evaluate exactly. $$\sin \left(-45^{\circ}\right) \sin \left(-15^{\circ}\right)$$

Evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. $$2 \sin \left(100^{\circ}\right) \sin \left(20^{\circ}\right)$$

Evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. $$2 \sin \left(-100^{\circ}\right) \sin \left(-20^{\circ}\right)$$

Evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine. $$\sin \left(213^{\circ}\right) \cos \left(8^{\circ}\right)$$