Is 2cos^2(θ)tan(θ)=sin(2θ) ?

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

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prove 2cos^2(θ)tan(θ)=sin(2θ)

prove

2 cos^{2} (θ) tan (θ) = sin (2 θ)

True

Solution steps

2 cos^{2} (θ) tan (θ) = sin (2 θ)

Manipulating left side

2 cos^{2} (θ) tan (θ)

Express with sin, cos

2 cos^{2} (θ) tan (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (θ)

= 2 sin (θ) cos (θ)

= 2 sin (θ) cos (θ)

= 2 cos (θ) sin (θ)

Rewrite using trig identities

2 cos (θ) sin (θ)

Use the Double Angle identity:

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

= sin (2 θ)

= sin (2 θ)

We showed that the two sides could take the same form

\Rightarrow True

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