Is 2cos^2(x)tan(x)=sin(2x) ?

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

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prove 2cos^2(x)tan(x)=sin(2x)

prove

2 cos^{2} (x) tan (x) = sin (2 x)

True

Solution steps

2 cos^{2} (x) tan (x) = sin (2 x)

Manipulating left side

2 cos^{2} (x) tan (x)

Express with sin, cos

2 cos^{2} (x) tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Cancel the common factor:

cos (x)

= 2 sin (x) cos (x)

= 2 sin (x) cos (x)

= 2 cos (x) sin (x)

Rewrite using trig identities

2 cos (x) sin (x)

Use the Double Angle identity:

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

= sin (2 x)

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

cos^{2} (θ) + cot^{2} (θ) cos^{2} (θ) = cot^{2} (θ)

\frac{sec ^{2} ( a )}{2 - sec ^{2} ( a )} = sec (2 a)

cot (a) + tan (a) = csc (a) sec (a)

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

sin (a + b) = sin (a) + sin (b)

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