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I need help with this question for my Precalculus class

I need help with this question for my Precalculus class-example-1
User Sergey Podobry
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1 Answer

19 votes
19 votes

If
x is positive, then
\tan^(-1)(x) is between 0 and π/2, which means
\cos(\tan^(-1)(x)) is between 1 and 0.

Let
\theta = \tan^(-1)(x). Then
\tan(\theta) = x, and it follows from the Pythagorean identity that


\sin^2(\theta) + \cos^2(\theta) = 1 \implies \tan^2(\theta) + 1 = \sec^2(\theta) \\\\ ~~~~ \implies \sec(\theta) = √(1 + \tan^2(\theta)) \\\\ ~~~~ \implies \cos(\theta) = \frac1{√(1+\tan^2(\theta))} \\\\ ~~~~ \implies \cos(\tan^(-1)(x)) = \boxed{\frac1{√(1+x^2)}}

We took the positive square root when solving for
\sec(\theta) because we know
\cos(\theta) is non-negative, between 0 and 1.

User MysticEarth
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