Is sin(2θ)=2cot(θ)sin^2(θ) ?

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

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prove sin(2θ)=2cot(θ)sin^2(θ)

prove

sin (2 θ) = 2 cot (θ) sin^{2} (θ)

True

Solution steps

sin (2 θ) = 2 cot (θ) sin^{2} (θ)

Manipulating right side

2 cot (θ) sin^{2} (θ)

Express with sin, cos

2 cot (θ) sin^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (θ)

= 2 cos (θ) sin (θ)

= 2 cos (θ) sin (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

tan (θ) + 1 = sec (θ)

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

cos (θ) - cos^{3} (θ) = sin^{2} (θ) cos (θ)

\frac{cos ( 4 x ) + 1}{sin ( 4 x )} = cot (2 x)

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

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