Is tan(2A)-2tan(2A)sin^2(A)=sin(2A) ?

The answer to whether tan(2A)-2tan(2A)sin^2(A)=sin(2A) is True

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prove tan(2A)-2tan(2A)sin^2(A)=sin(2A)

prove

tan (2 A) - 2 tan (2 A) sin^{2} (A) = sin (2 A)

True

Solution steps

tan (2 A) - 2 tan (2 A) sin^{2} (A) = sin (2 A)

Manipulating left side

tan (2 A) - 2 tan (2 A) sin^{2} (A)

Express with sin, cos

tan (2 A) - 2 sin^{2} (A) tan (2 A)

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{sin ( 2 A )}{cos ( 2 A )} - 2 sin^{2} (A) \frac{sin ( 2 A )}{cos ( 2 A )}

Multiply

2 sin^{2} (A) \frac{sin ( 2 A )}{cos ( 2 A )} : \frac{2 sin ^{2} ( A ) sin ( 2 A )}{cos ( 2 A )}

2 sin^{2} (A) \frac{sin ( 2 A )}{cos ( 2 A )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( 2 A ) \cdot 2 sin ^{2} ( A )}{cos ( 2 A )}

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

Rewrite using trig identities

\frac{sin ( 2 A ) - 2 sin ( 2 A ) sin ^{2} ( A )}{cos ( 2 A )}

Use the Double Angle identity:

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

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

= \frac{sin ( 2 A ) + ( cos ( 2 A ) - 1 ) sin ( 2 A )}{cos ( 2 A )}

Simplify

\frac{sin ( 2 A ) + ( cos ( 2 A ) - 1 ) sin ( 2 A )}{cos ( 2 A )} : sin (2 A)

\frac{sin ( 2 A ) + ( cos ( 2 A ) - 1 ) sin ( 2 A )}{cos ( 2 A )}

Factor

sin (2 A) + (cos (2 A) - 1) sin (2 A) : sin (2 A) cos (2 A)

sin (2 A) + (cos (2 A) - 1) sin (2 A)

Factor out common term

sin (2 A)

= sin (2 A) (1 - 1 + cos (2 A))

Refine

= sin (2 A) cos (2 A)

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

Cancel the common factor:

cos (2 A)

= sin (2 A)

= sin (2 A)

= sin (2 A)

We showed that the two sides could take the same form

\Rightarrow True

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