91Ó°ÊÓ

Verify each of the trigonometric identities. $$\csc (-x)-1=\frac{\cot ^{2} x}{\csc (-x)+1}$$

Use the half-angle identities to find the exact values of the trigonometric expressions. $$\sin 15^{\circ}$$

Solve the given trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ}\) and express the answer in degrees to two decimal places. $$6 \tan ^{2} \theta-\tan \theta-12=0$$

Explain the mistake that is made. Simplify the expression \((\cos A-\cos B)^{2}+(\sin A-\sin B)^{2}\) Solution: Expand by squaring. \(\cos ^{2} A-2 \cos A \cos B+\cos ^{2} B+\sin ^{2} A-2 \sin A \sin B+\sin ^{2} B\) Group terms. \(\cos ^{2} A+\sin ^{2} A-2 \cos A \cos B-2 \sin A \sin B+\cos ^{2} B+\sin ^{2} B\) Simplify using the Pythagorean identity. \(\underbrace{\cos ^{2} A+\sin ^{2} A}_{1}-2 \cos A \cos B-2 \sin A \sin B+\underbrace{\cos ^{2} B+\sin ^{2} B}_{1}\) Factor the common 2 \(2(1-\cos A \cos B-\sin A \sin B)\) Simplify. \(2(1-\cos A B-\sin A B)\) This is incorrect. What mistakes were made?

Determine whether each equation is a conditional equation or an identity. $$\frac{1+\tan x}{1-\tan x}=\tan \left(x-\frac{\pi}{4}\right)$$

Verify each of the trigonometric identities. $$\frac{1}{\csc ^{2} x}+\frac{1}{\sec ^{2} x}=1$$

Verify each of the trigonometric identities. $$\frac{1}{\cot ^{2} x}-\frac{1}{\tan ^{2} x}=\sec ^{2} x-\csc ^{2} x$$

Solve the given trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ}\) and express the answer in degrees to two decimal places. $$6 \sec ^{2} \theta-7 \sec \theta-20=0$$

Determine whether each equation is a conditional equation or an identity. $$\cot \left(x+\frac{\pi}{4}\right)=\frac{1-\tan x}{1+\tan x}$$

Use the half-angle identities to find the exact values of the trigonometric expressions. $$\cos 22.5^{\circ}$$