Is (sin(2A))/(1-cos(2A))=cot(A) ?

The answer to whether (sin(2A))/(1-cos(2A))=cot(A) is True

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prove (sin(2A))/(1-cos(2A))=cot(A)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

sin (A)

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

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

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

Use the basic trigonometric identity:

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

= cot (A)

We showed that the two sides could take the same form

\Rightarrow True

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