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(2tan(x))/(1+tan^(2)(x))=sen(2x)

$\frac{2 \tan (x)}{1+\tan ^{2}(x)}=\operatorname{sen}(2 x)$

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Evaluate integers raised to rational exponents

Full solution

Q.

\frac{2 \tan (x)}{1+\tan ^{2}(x)}=\operatorname{sen}(2 x)

Apply tan identity: Use the identity $\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$ and the Pythagorean identity $\tan^2(x) + 1 = \sec^2(x)$ .
Use Pythagorean identity: Rewrite the given expression using the Pythagorean identity: $\frac{2\tan(x)}{\sec^2(x)}$ .
Rewrite using secant: Since $\sec(x) = \frac{1}{\cos(x)}$ , rewrite $\sec^2(x)$ as $\frac{1}{\cos^2(x)}$ .
Simplify expression: Now the expression is $(2\tan(x)) \cdot \cos^2(x)$ .
Use sin identity: Use the identity $\sin(2x) = 2\sin(x)\cos(x)$ and divide both sides by $\cos(x)$ to get $\tan(x) = \frac{\sin(x)}{\cos(x)}$ .
Substitute tan: Substitute $\tan(x)$ with $\frac{\sin(x)}{\cos(x)}$ in the expression: $2\left(\frac{\sin(x)}{\cos(x)}\right) \cdot \cos^2(x)$ .
Final simplification: Simplify the expression: $2\sin(x)\cos(x)$ .
Recognize pattern: Recognize that $2\sin(x)\cos(x)$ is the right side of the double angle identity for sine, which is $\sin(2x)$ .
Conclude identity: Conclude that $(2\tan(x))/(1+\tan^{2}(x))$ simplifies to $\sin(2x)$ , proving the identity.

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