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Magazine Problem 81
In Exercises \(69-82,\) prove the given identities. $$-\sin (x+y) \sin (x-y)=\sin ^{2} x \cos ^{2} y-\cos ^{2} x \sin ^{2} y$$
Problem 81
Find the exact value of each expression. $$\sin ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{4}{5}\right)$$
Problem 81
In Exercises \(81-86,\) use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities. $$\sin (x+\pi)=\sin x+\pi$$
Problem 82
Find the exact value of each expression. $$\cos ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{1}{2}\right)$$
Problem 82
In Exercises \(69-82,\) prove the given identities. $$\cos (x-y) \cos (x+y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$
Problem 82
Use a scientific calculator to find the solutions of the given equations, in radians, that lie in the interval \([0,2 \pi)\). $$10 \tan ^{4} x-7 \sec ^{4} x+23=0$$
Problem 82
In Exercises \(81-86,\) use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities. $$\cos 2 x=2 \cos x$$
Problem 83
Use a graphing utility to find the solutions of the given equations, in radians, that lie in the interval \([0,2 \pi)\). $$\cos 2 x=-x^{2}$$
Problem 83
In Exercises \(83-88,\) find the exact value of each expression. $$\sin \left(\cos ^{-1} 0-\sin ^{-1} \frac{1}{2}\right)$$
Problem 83
In Exercises \(81-86,\) use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities. $$\cos 2 x=1-2 \sin ^{2} x$$
