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.cot² θ(1 + tan² θ)

Accepted Answer

Recall the definitions of cotangent and tangent in terms of sine and cosine: $$\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$$ and $$	an 	heta = \frac{\sin 	heta}{\cos 	heta}. $$Rewrite the given expression $$\cot$$^{2}$$ 	heta (1 + 	an$$^{2}$$ 	heta) by $$substituting these definitions: $$\left(\frac{\cos 	heta}{\sin 	heta}\$$right)^{2}$$ \left(1 + \left(\frac{\sin 	heta}{\cos 	heta}\$$right)^{2}$$ight). $$Simplify inside the parentheses first: $$1 + \left(\frac{\sin 	heta}{\cos 	heta}\$$right)^{2}$$ = 1 + \frac{\sin$$^{2}$$ 	heta}{\cos$$^{2}$$ 	heta}. $$Combine the terms inside the parentheses over a common denominator: $$\frac{\cos$$^{2}$$ 	heta}{\cos$$^{2}$$ 	heta} + \frac{\sin$$^{2}$$ 	heta}{\cos$$^{2}$$ 	heta} = \frac{\cos$$^{2}$$ 	heta + \sin$$^{2}$$ 	heta}{\cos$$^{2}$$ 	heta}. $$Use the Pythagorean identity $$\sin$$^{2}$$ 	heta + \cos$$^{2}$$ 	heta = 1 to $$simplify the numerator, then multiply by the squared cotangent term and simplify the entire expression so that no quotients remain and all functions are in terms of $$	heta$$ only.

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.
cot² θ(1 + tan² θ)

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Key Concepts

Trigonometric Ratios and Their Definitions

Pythagorean Identity

Algebraic Simplification of Trigonometric Expressions