Is (1-cos(θ))(1+cos(θ))=sin^2(θ) ?

The answer to whether (1-cos(θ))(1+cos(θ))=sin^2(θ) is True

prove (1-cos(θ))(1+cos(θ))=sin^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

(1 - cos (θ)) (1 + cos (θ))

Expand

(1 + cos (θ)) (1 - cos (θ)) : 1 - cos^{2} (θ)

(1 + cos (θ)) (1 - cos (θ))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = cos (θ)

= 1^{2} - cos^{2} (θ)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - cos^{2} (θ)

= 1 - cos^{2} (θ)

Use the Pythagorean identity:

1 = cos^{2} (θ) + sin^{2} (θ)

1 - cos^{2} (θ) = sin^{2} (θ)

= sin^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{1 - 3 cos ( x ) - 4 cos ^{2} ( x )}{sin ^{2} ( x )} = \frac{1 - 4 cos ( x )}{1 - cos ( x )}

p ro v e (1 + cos (2 x)) (1 - cos (2 x)) = sin^{2} (2 x)

p ro v e \frac{1}{sin ( x ) + 1} + \frac{1}{csc ( x ) + 1} = 1

p ro v e sin (t) csc (t) = 1

p ro v e cos (x) = sin (x + \frac{π}{2})

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