Is (sin(2x))/(cos(2x)+sin^2(x))=2tan(x) ?

The answer to whether (sin(2x))/(cos(2x)+sin^2(x))=2tan(x) is True

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prove (sin(2x))/(cos(2x)+sin^2(x))=2tan(x)

prove

\frac{sin ( 2 x )}{cos ( 2 x ) + sin ^{2} ( x )} = 2 tan (x)

True

Solution steps

\frac{sin ( 2 x )}{cos ( 2 x ) + sin ^{2} ( x )} = 2 tan (x)

Manipulating left side

\frac{sin ( 2 x )}{cos ( 2 x ) + sin ^{2} ( x )}

Rewrite using trig identities

\frac{sin ( 2 x )}{cos ( 2 x ) + sin ^{2} ( x )}

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= \frac{2 sin ( x ) cos ( x )}{cos ( 2 x ) + sin ^{2} ( x )}

Use the Double Angle identity:

cos (2 x) = cos^{2} (x) - sin^{2} (x)

= \frac{2 sin ( x ) cos ( x )}{cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{2} ( x )}

Simplify

\frac{2 sin ( x ) cos ( x )}{cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{2} ( x )} : \frac{2 sin ( x )}{cos ( x )}

\frac{2 sin ( x ) cos ( x )}{cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{2} ( x )}

Add similar elements:

- sin^{2} (x) + sin^{2} (x) = 0

= \frac{2 sin ( x ) cos ( x )}{cos ^{2} ( x )}

Cancel the common factor:

cos (x)

= \frac{2 sin ( x )}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )}

Rewrite using trig identities

= 2 \cdot \frac{sin ( x )}{cos ( x )}

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

2 tan (x)

2 tan (x)

We showed that the two sides could take the same form

\Rightarrow True

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Is (sin(2x))/(cos(2x)+sin^2(x))=2tan(x) ?
The answer to whether (sin(2x))/(cos(2x)+sin^2(x))=2tan(x) is True