91Ó°ÊÓ

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. $$2 \cos (2 x) \cos x+\sin (2 x) \sin x=2 \sin x$$

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. $$\frac{\sin (2 x)+\sin (4 x)}{\sin (2 x)-\sin (4 x)}=-\tan (3 x) \cot x$$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. $$\frac{\sin (9 t)-\sin (3 t)}{\cos (9 t)+\cos (3 t)}$$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. $$2 \sin (8 x) \cos (6 x)-\sin (2 x)$$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. $$\frac{\sin (3 x)-\sin x}{\sin x}$$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. $$\frac{\cos (5 x)+\cos (3 x)}{\sin (5 x)+\sin (3 x)}$$

Simplify the expression to one term, then graph the original function and your simplified version to verify they are identical. $$\sin x \cos (15 x)-\cos x \sin (15 x)$$

Prove the following sum-to-product formulas. $$\sin x-\sin y=2 \sin \left(\frac{x-y}{2}\right) \cos \left(\frac{x+y}{2}\right)$$

Prove the following sum-to-product formulas. $$\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$

Prove the identity. $$\frac{\sin (6 x)+\sin (4 x)}{\sin (6 x)-\sin (4 x)}=\tan (5 x) \cot x$$