Is cot(x)(1-cos(2x))=sin(2x) ?

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

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prove cot(x)(1-cos(2x))=sin(2x)

prove

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

True

Solution steps

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

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

tan (θ) cot (θ) = 1

cos (\frac{π}{2} - x) = sin (x)

sin (\frac{π}{2} - θ) = cos (θ)

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

sec (u) - tan (u) = \frac{cos ( u )}{1 + sin ( u )}

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