Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.-sec² (-θ) + sin² (-θ) + cos² (-θ)

Accepted Answer

Recall the definitions and identities for the trigonometric functions involved: $$\sec 	heta = \frac{1}{\cos 	heta}$$, so $$\sec^2$ 	heta = \frac{1}{\cos^2$ 	heta}. $$Also, use the even-odd properties: $$\cos(-	heta) = \cos 	heta ($$cosine is even) and $$\sin(-	heta) = -\sin 	heta ($$sine is odd). Rewrite each term in the expression $$-\sec^2$(-	heta) + \sin^2$(-	heta) + \cos^2$(-	heta)$$ using sine and cosine functions and apply the even-odd properties: $$-\sec^2$(-	heta) = -\frac{1}{\cos^2$(-	heta)} = -\frac{1}{\cos^2$ 	heta}$$, $$\sin^2$(-	heta) = (-\sin \$$theta)^2$$ = \sin^2$ 	heta$$, and $$\cos^2$(-	heta) = \cos^2$ 	heta. $$Substitute these back into the expression to get $$-\frac{1}{\cos^2$ 	heta} + \sin^2$ 	heta + \cos^2$ 	heta. $$Use the Pythagorean identity $$\sin^2$ 	heta + \cos^2$ 	heta = 1 to $$simplify the expression to $$-\frac{1}{\cos^2$ 	heta} + 1. $$Rewrite $$\frac{1}{\cos^2$ 	heta} as$$ $$\sec^2$ 	heta if $$needed, or multiply through by $$\cos^2$ 	heta to $$eliminate the quotient, depending on the instruction to avoid quotients, and simplify accordingly.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
-sec² (-θ) + sin² (-θ) + cos² (-θ)

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Key Concepts

Even and Odd Trigonometric Functions

Trigonometric Identities

Expressing Trigonometric Functions in Terms of Sine and Cosine