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

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

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

prove

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

True

Solution steps

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

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

Simplify

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

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

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

1 + 2 cos^{2} (a) - 1

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (a)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (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

tan (x) = tan (x + π)

1 + csc (a) = csc (a) (1 + sin (a))

1 + cot (θ) = csc (θ) (cos (θ) + sin (θ))

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

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

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