Is (sin(2a))/(1-sin^2(a))=2tan(a) ?

The answer to whether (sin(2a))/(1-sin^2(a))=2tan(a) is True

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prove (sin(2a))/(1-sin^2(a))=2tan(a)

prove

\frac{sin ( 2 a )}{1 - sin ^{2} ( a )} = 2 tan (a)

True

Solution steps

\frac{sin ( 2 a )}{1 - sin ^{2} ( a )} = 2 tan (a)

Manipulating left side

\frac{sin ( 2 a )}{1 - sin ^{2} ( a )}

Rewrite using trig identities

\frac{sin ( 2 a )}{1 - sin ^{2} ( a )}

Use the Double Angle identity:

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

= \frac{2 sin ( a ) cos ( a )}{1 - sin ^{2} ( a )}

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

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

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

Cancel the common factor:

cos (a)

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

2 tan (a)

2 tan (a)

We showed that the two sides could take the same form

\Rightarrow True

Is (sin(2a))/(1-sin^2(a))=2tan(a) ?
The answer to whether (sin(2a))/(1-sin^2(a))=2tan(a) is True