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.[1 - sin²(-θ)]/[1 + cot²(-θ)]

Accepted Answer

Recall the Pythagorean identity: $$\sin^2$(	heta) + \cos^2$(	heta) = 1. $$This will help simplify expressions involving $$1 - \sin^2$(	heta). $$Rewrite the numerator $$1 - \sin^2$(-	heta)$$ using the identity. Since $$\sin(-	heta) = -\sin(	heta)$$, we have $$\sin^2$(-	heta) = \sin^2$(	heta)$$, so the numerator becomes $$1 - \sin^2$(	heta). $$Rewrite the denominator $$1 + \cot^2$(-	heta). $$Recall that $$\cot(	heta) = \frac{\cos(	heta)}{\sin(	heta)}$$ and $$\cot(-	heta) = -\cot(	heta)$$, so $$\cot^2$(-	heta) = \cot^2$(	heta). $$Use the Pythagorean identity $$1 + \cot^2$(	heta) = \csc^2$(	heta) to $$rewrite the denominator. Express $$\csc(	heta) in $$terms of sine: $$\csc(	heta) = \frac{1}{\sin(	heta)}$$, so $$\csc^2$(	heta) = \frac{1}{\sin^2$(	heta)}. $$Substitute this into the denominator. Combine the simplified numerator and denominator: $$\frac{1 - \sin^2$(	heta)}{\frac{1}{\sin^2$(	heta)}}. $$Then rewrite $$1 - \sin^2$(	heta) as$$ $$\cos^2$(	heta)$$ and multiply by the reciprocal of the denominator to eliminate the fraction, resulting in an expression involving only sine and cosine with no quotients.

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.
[1 - sin²(-θ)]/[1 + cot²(-θ)]

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Key Concepts

Pythagorean Identities

Even-Odd Properties of Trigonometric Functions

Definition and Conversion of Cotangent