Express (1-tan^(2)theta)/(1+tan^(2)theta) in terms of sin theta.

Rewrite using Pythagorean identity: Use the Pythagorean identity tan^2(theta) = sin^2(theta)/cos^2(theta) to rewrite the expression. (1-tan^(2)theta)/(1+tan^(2)theta) = (1-(sin^2(theta)/cos^2(theta)))/(1+(sin^2(theta)/cos^2(theta)))Combine terms over common denominator: Combine the terms over a common denominator. = (cos^2(theta)-sin^2(theta))/(cos^2(theta)+sin^2(theta))Recognize denominator identity: Recognize that the denominator is another Pythagorean identity: cos^2(theta)+sin^2(theta) = 1. = (cos^2(theta)-sin^2(theta))/1Simplify the expression: Simplify the expression. = cos^2(theta)-sin^2(theta)Express cos^2(theta) in terms of sin^2(theta): Use the Pythagorean identity sin^2(theta) = 1 - cos^2(theta) to express cos^2(theta) in terms of sin^2(theta). = (1-sin^2(theta))-sin^2(theta)Combine like terms: Combine like terms. = 1 - 2sin^2(theta)

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Express
(1-tan^(2)theta)/(1+tan^(2)theta) in terms of
sin theta.

Express $\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}$ in terms of $\sin \theta$ .

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Calculus

Evaluate definite integrals using the chain rule

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Q. Express

\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}

in terms of

\sin \theta

Rewrite using Pythagorean identity: Use the Pythagorean identity $\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}$ to rewrite the expression. $\newline$ $\frac{1-\tan^2(\theta)}{1+\tan^2(\theta)} = \frac{1-\left(\frac{\sin^2(\theta)}{\cos^2(\theta)}\right)}{1+\left(\frac{\sin^2(\theta)}{\cos^2(\theta)}\right)}$
Combine terms over common denominator: Combine the terms over a common denominator. $\newline$ = $\frac{\cos^2(\theta)-\sin^2(\theta)}{\cos^2(\theta)+\sin^2(\theta)}$
Recognize denominator identity: Recognize that the denominator is another Pythagorean identity: $\cos^2(\theta)+\sin^2(\theta) = 1$ .
= $(\cos^2(\theta)-\sin^2(\theta))/1$
Simplify the expression: Simplify the expression. $\newline$ $= \cos^2(\theta)-\sin^2(\theta)$
Express $\cos^2(\theta)$ in terms of $\sin^2(\theta)$ : Use the Pythagorean identity $\sin^2(\theta) = 1 - \cos^2(\theta)$ to express $\cos^2(\theta)$ in terms of $\sin^2(\theta)$ . $\newline$ = $(1-\sin^2(\theta))-\sin^2(\theta)$
Combine like terms: Combine like terms. $\newline$ $= 1 - 2\sin^2(\theta)$