Question

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.csc² θ + sec² θ

Accepted Answer

Recall the definitions of the cosecant and secant functions in terms of sine and cosine: $$\csc 	heta = \frac{1}{\sin 	heta}$$ and $$\sec 	heta = \frac{1}{\cos 	heta}. $$Rewrite the given expression $$\csc$$^{2}$$ 	heta + \sec$$^{2}$$ 	heta$$ using these definitions: $$\left(\frac{1}{\sin 	heta}\$$right)^{2}$$ + \left(\frac{1}{\cos 	heta}\$$right)^{2}$$. Simplify the expression to $$\frac{1}{\$$sin^{2}$$ 	heta} + \frac{1}{\$$cos^{2}$$ 	heta}$$, which is a sum of two fractions. Find a common denominator for the two fractions, which is $$\$$sin^{2}$$ 	heta \$$cos^{2}$$ 	heta$$, and rewrite the expression as $$\frac{\$$cos^{2}$$ 	heta}{\$$sin^{2}$$ 	heta \$$cos^{2}$$ 	heta} + \frac{\$$sin^{2}$$ 	heta}{\$$sin^{2}$$ 	heta \$$cos^{2}$$ 	heta}$$. Combine the fractions into a single fraction: $$\frac{\$$cos^{2}$$ 	heta + \$$sin^{2}$$ 	heta}{\$$sin^{2}$$ 	heta \$$cos^{2}$$ 	heta}$$, then use the Pythagorean identity $$\$$sin^{2}$$ 	heta + \$$cos^{2}$$ 	heta = 1$$ to simplify the numerator.

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc² θ + sec² θ

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Key Concepts

Reciprocal Trigonometric Identities

Pythagorean Identities

Algebraic Simplification of Trigonometric Expressions