91Ó°ÊÓ

Simplify the given expressions. The result will be one of \(\sin x, \cos x,\) tan \(x, \cot x, \sec x,\) or \(\csc x\). $$\sin x(\tan x+\cot x)$$

Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. $$\frac{1-\tan ^{2} x}{\sec ^{2} x}=\cos 2 x$$

Solve the given problems. Show that \(\frac{\sin 2 x}{\sin x}=2 \cos x\). \([\text { Hint: } \sin 2 x=\sin (x+x) .]\)

Use the half-angle formulas to solve the given problems. In finding the path of a sliding particle, the expression \(\sqrt{8-8 \cos \theta}\) is used. Simplify this expression.

Simplify the given expressions. The result will be one of \(\sin x, \cos x,\) tan \(x, \cot x, \sec x,\) or \(\csc x\). $$\frac{\tan x+\cot x}{\csc x}$$

Solve the indicated equations analytically. Solve the system of equations \(r=\sin \theta, r=\sin 2 \theta,\) for \(0 \leq \theta<2 \pi\).

Find an algebraic expression for each of the given expressions. $$\tan \left(\sin ^{-1} x\right)$$

Use the half-angle formulas to solve the given problems. In designing track for a railway system, the equation \(d=4 r \sin ^{2} \frac{A}{2}\) is used. Solve for \(d\) in terms of \(\cos A\).

Solve the indicated equations analytically. Solve the system of equations \(r=\sin \theta, r=\cos 2 \theta,\) for \(0 \leq \theta<2 \pi\).

Find an algebraic expression for each of the given expressions. $$\sin \left(\cos ^{-1} x\right)$$