Prove the identity. sin(2x) 2 tan(x) 1 + tan²(x) Write the more complicated side of the equation in terms of sin(x) and cos(x). Simplify the complex fraction. sin(x) 2 tan(x) 1 + tan²(x) = 2. 1 + 2 1+ 2. sin(x) sin(x) cos(x) sin²(x) cos²(x) cos(x) sin (2x) cos²(x) cos²(x) + sin²(x) 2 sin(x) cos(x) cos²(x) sin²(x) X
Prove the identity. sin(2x) 2 tan(x) 1 + tan²(x) Write the more complicated side of the equation in terms of sin(x) and cos(x). Simplify the complex fraction. sin(x) 2 tan(x) 1 + tan²(x) = 2. 1 + 2 1+ 2. sin(x) sin(x) cos(x) sin²(x) cos²(x) cos(x) sin (2x) cos²(x) cos²(x) + sin²(x) 2 sin(x) cos(x) cos²(x) sin²(x) X
Transcribed Image Text:Prove the identity.
\[
\frac{\sin(2x)}{1 + \tan^2(x)} = \frac{2 \tan(x)}{1 + \tan^2(x)}
\]
Write the more complicated side of the equation in terms of \(\sin(x)\) and \(\cos(x)\). Simplify the complex fraction.
\[
\frac{2 \tan(x)}{1 + \tan^2(x)} = \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin(x)}{\cos^2(x)}}
\]
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin^2(x)}{\cos^2(x)}}
\]
Marked with a green check (\(\sin(x)/\cos(x)\)) indicates correct steps, while red crosses indicate incorrect placements or notations.
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{\cos^2(x)/\sin^2(x)}
\]
\[
= \frac{2 \cdot \sin(x) \cdot \cos(x)}{\cos^2(x) + \sin^2(x)}
\]
\[
= \frac{2 \sin(x) \cos(x)}{1}
\]
\[
= \sin(2x)
\]
In this solution, the identity for \(\sin(2x) = 2 \sin(x) \cos(x)\) is used to simplify and confirm the trigonometric identity.
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![Prove the identity.
\[
\frac{\sin(2x)}{1 + \tan^2(x)} = \frac{2 \tan(x)}{1 + \tan^2(x)}
\]
Write the more complicated side of the equation in terms of \(\sin(x)\) and \(\cos(x)\). Simplify the complex fraction.
\[
\frac{2 \tan(x)}{1 + \tan^2(x)} = \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin(x)}{\cos^2(x)}}
\]
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{1 + \frac{\sin^2(x)}{\cos^2(x)}}
\]
Marked with a green check (\(\sin(x)/\cos(x)\)) indicates correct steps, while red crosses indicate incorrect placements or notations.
\[
= \frac{2 \cdot \frac{\sin(x)}{\cos(x)}}{\cos^2(x)/\sin^2(x)}
\]
\[
= \frac{2 \cdot \sin(x) \cdot \cos(x)}{\cos^2(x) + \sin^2(x)}
\]
\[
= \frac{2 \sin(x) \cos(x)}{1}
\]
\[
= \sin(2x)
\]
In this solution, the identity for \(\sin(2x) = 2 \sin(x) \cos(x)\) is used to simplify and confirm the trigonometric identity.](https://content.bartleby.com/qna-images/question/8e308213-df48-4360-8e9d-bfd0a573e9aa/0f2a8709-c340-45ef-9b52-9d271d4dbd0b/qorwut4_processed.png)
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