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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

cos (x)

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

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

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

Use the basic trigonometric identity:

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

= tan (x)

We showed that the two sides could take the same form

\Rightarrow True

cos (3 x) + cos (x) = 2 cos (2 x) cos (x)

sin (4 x) = 2 sin (2 x) cos (2 x)

tan (x) csc^{2} (x) - tan (x) = cot (x)

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

sin (θ) + cos (θ) cot (θ) = csc (θ)

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