Is 2sin(θ)cos(θ)=tan(2θ)cos(2θ) ?

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

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prove 2sin(θ)cos(θ)=tan(2θ)cos(2θ)

prove

2 sin (θ) cos (θ) = tan (2 θ) cos (2 θ)

True

Solution steps

2 sin (θ) cos (θ) = tan (2 θ) cos (2 θ)

Manipulating right side

tan (2 θ) cos (2 θ)

Express with sin, cos

cos (2 θ) tan (2 θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos (2 θ)

= sin (2 θ)

= sin (2 θ)

= sin (2 θ)

Rewrite using trig identities

sin (2 θ)

Use the Double Angle identity:

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

= 2 sin (θ) cos (θ)

= 2 sin (θ) cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

cos (θ) sec (θ) - cos^{2} (θ) = sin^{2} (θ)

sin (12 x) = 2 sin (6 x) cos (6 x)

\frac{sin ( t )}{csc ( t )} = 1 - \frac{cos ( t )}{sec ( t )}

cot^{2} (x) = 1 - csc^{2} (x)

tan^{4} (θ) = tan^{2} (θ) sec^{2} (θ) - tan^{2} (θ)

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