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

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

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prove tan(2x)=(2sin(x)cos(x))/(2cos^2(x)-1)

prove

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

True

Solution steps

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

Manipulating left side

tan (2 x)

Express with sin, cos

tan (2 x)

Use the basic trigonometric identity:

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

tan (\frac{x}{2}) = - cot (x) + csc (x)

\frac{1}{csc ^{2} ( x )} + \frac{1}{sec ^{2} ( x )} = 1

cot (\frac{5 π}{12}) = 2 - 3

cos (4 x) = cos^{2} (2 x) - sin^{2} (2 x)

\frac{cos ^{3} ( t )}{cot ^{2} ( t )} = sin^{2} (t) cos (t)

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