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

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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

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

\frac{tan ( θ )}{sec ( θ )} = sin (θ)

\frac{tan ( x )}{sin ( x )} = sec (x)

(1 - sin^{2} (θ)) sec^{2} (θ) = 1

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

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