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

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

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prove cot(x)(1-cos(2x))=2cos^2(x)tan(x)

prove

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

True

Solution steps

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

Manipulating left side

cot (x) (1 - cos (2 x))

Rewrite using trig identities

cot (x) (1 - cos (2 x))

Use the Double Angle identity:

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

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

Expand

1 - (1 - 2 sin^{2} (x)) : 2 sin^{2} (x)

1 - (1 - 2 sin^{2} (x))

- (1 - 2 sin^{2} (x)) : - 1 + 2 sin^{2} (x)

- (1 - 2 sin^{2} (x))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + 2 sin^{2} (x)

= 1 - 1 + 2 sin^{2} (x)

1 - 1 = 0

= 2 sin^{2} (x)

= 2 sin^{2} (x) cot (x)

= 2 sin^{2} (x) cot (x)

Express with sin, cos

2 cot (x) sin^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

= 2 cos (x) sin (x)

Manipulating right 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)

We showed that the two sides could take the same form

\Rightarrow True

8 \cdot sin^{4} (θ) = cos (4 θ) + 4 \cdot cos (2 θ) - 3

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

tan^{2} (θ) = sin^{2} (θ) sec^{2} (θ)

cos^{2} (θ) - sin^{2} (θ) = 1 - tan^{2} (θ)

tan (6 x) sec (3 x) = 2 sin (3 x) sec (6 x)

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