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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

Expand

1 - (1 - 2 sin^{2} (a)) : 2 sin^{2} (a)

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

- (1 - 2 sin^{2} (a)) : - 1 + 2 sin^{2} (a)

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

Distribute parentheses

= - (1) - (- 2 sin^{2} (a))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + 2 sin^{2} (a)

= 1 - 1 + 2 sin^{2} (a)

1 - 1 = 0

= 2 sin^{2} (a)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

sin (a)

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

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

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

Use the basic trigonometric identity:

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

= tan (a)

We showed that the two sides could take the same form

\Rightarrow True

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