Is (sin(2α))/(1+cos(2α))=tan(α) ?

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

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prove (sin(2α))/(1+cos(2α))=tan(α)

prove

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

True

Solution steps

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

Manipulating left side

\frac{sin ( 2 α )}{1 + cos ( 2 α )}

Rewrite using trig identities

\frac{sin ( 2 α )}{1 + cos ( 2 α )}

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

1 + 2 cos^{2} (α) - 1 = 2 cos^{2} (α)

1 + 2 cos^{2} (α) - 1

Group like terms

= 2 cos^{2} (α) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (α)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (α)

= \frac{sin ( α )}{cos ( α )}

= \frac{sin ( α )}{cos ( α )}

= \frac{sin ( α )}{cos ( α )}

Use the basic trigonometric identity:

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

= tan (α)

We showed that the two sides could take the same form

\Rightarrow True

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