91Ó°ÊÓ

Show that \(\cos 87^{\circ}+\cos 33^{\circ}=\sin 63^{\circ}\)

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that \(0 \leq \theta<\pi / 2\) $$\frac{1}{x^{2} \sqrt{4+x^{2}}}, \quad x=2 \tan \theta$$

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that \(0 \leq \theta<\pi / 2\) $$\sqrt{9-x^{2}}, \quad x=3 \sin \theta$$

Prove the identity $$\frac{\sin x+\sin 2 x+\sin 3 x+\sin 4 x+\sin 5 x}{\cos x+\cos 2 x+\cos 3 x+\cos 4 x+\cos 5 x}=\tan 3 x$$

Use the identity $$\sin 2 x=2 \sin x \cos x$$ \(n\) times to show that \(\sin \left(2^{n} x\right)=2^{n} \sin x \cos x \cos 2 x \cos 4 x \cdots \cos 2^{n-1} x\)

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that \(0 \leq \theta<\pi / 2\) $$\frac{\sqrt{x^{2}-25}}{x}, \quad x=5 \sec \theta$$

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=\cos ^{2} x-\sin ^{2} x, \quad g(x)=1-2 \sin ^{2} x$$

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=\tan x(1+\sin x), \quad g(x)=\frac{\sin x \cos x}{1+\sin x}$$

(a) Graph \(f(x)=\cos 2 x+2 \sin ^{2} x\) and make a conjecture. (b) Prove the conjecture you made in part (a).

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=(\sin x+\cos x)^{2}, \quad g(x)=1$$