91Ó°ÊÓ

Verify that each trigonometric equation is an identity. $$\tan ^{2} \alpha \sin ^{2} \alpha=\tan ^{2} \alpha+\cos ^{2} \alpha-1$$

Use identities to find each exact value. $$\sin 100^{\circ} \cos 10^{\circ}-\cos 100^{\circ} \sin 10^{\circ}$$

Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of \(\theta\) only. $$\csc \theta \cos \theta \tan \theta$$

Verify that each trigonometric equation is an identity. $$\frac{\tan x}{1+\cos x}+\frac{\sin x}{1-\cos x}=\cot x+\sec x \csc x$$

Find each of the following. $$\tan \frac{\theta}{2}, \text { given } \tan \theta=\frac{\sqrt{7}}{3}, \text { with } 180^{\circ} < \theta < 270^{\circ}$$

Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of \(\theta\) only. $$\cos \theta \csc \theta$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right),\) and your work leads to \(3 \theta=180^{\circ}, 630^{\circ}, 720^{\circ}, 930^{\circ} .\) What are the cor- responding values of \(\theta ?\)

Find each of the following. $$\cot \frac{\theta}{2}, \text { given } \tan \theta=-\frac{\sqrt{5}}{2}, \text { with } 90^{\circ} < \theta < 180^{\circ}$$

Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of \(\theta\) only. $$\sin \theta \sec \theta$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right),\) and your work leads to \(\frac{1}{3} \theta=45^{\circ}, 60^{\circ}, 75^{\circ}, 90^{\circ} .\) What are the corresponding values of \(\theta ?\)