Is 2cot^2(x)sin^2(x)=1+cos(2x) ?

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

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prove 2cot^2(x)sin^2(x)=1+cos(2x)

prove

2 cot^{2} (x) sin^{2} (x) = 1 + cos (2 x)

True

Solution steps

2 cot^{2} (x) sin^{2} (x) = 1 + cos (2 x)

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{2} (x)

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

= 2 cos^{2} (x)

= 2 cos^{2} (x)

Rewrite using trig identities

2 cos^{2} (x)

Use the Pythagorean identity:

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

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

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

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

Expand

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

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Rewrite using trig identities

2 - 2 sin^{2} (x)

Use the Double Angle identity:

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

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

= 2 + cos (2 x) - 1

Simplify

2 + cos (2 x) - 1 : cos (2 x) + 1

2 + cos (2 x) - 1

Group like terms

= cos (2 x) + 2 - 1

Add/Subtract the numbers:

2 - 1 = 1

= cos (2 x) + 1

= cos (2 x) + 1

= cos (2 x) + 1

= 1 + cos (2 x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{tan ( x )} - tan (x) = 2 cot (2 x)

\frac{csc ^{2} ( θ ) - 1}{csc ^{2} ( θ )} = cos^{2} (θ)

1 + cos (x) \cdot sin (x) = sin (x)

(sin (x)) \frac{1}{sin ( x )} = 1

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

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