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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(1 + sin (x)) (1 + sin (- x))

Use the negative angle identity:

sin (- x) = - sin (x)

= (1 - sin (x)) (1 + sin (x))

Rewrite using trig identities

(1 - sin (x)) (1 + sin (x))

Expand

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

(1 + sin (x)) (1 - sin (x))

Apply Difference of Two Squares Formula:

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

a = 1, b = sin (x)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (x)

= 1 - sin^{2} (x)

Use the Pythagorean identity:

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

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

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

1 - tan^{2} (\frac{x}{2}) = \frac{2 cos ( x )}{1 + cos ( x )}

\frac{1 + cos ( θ )}{1 + sec ( θ )} = cos (θ)

tan^{2} (θ) - sec^{2} (θ) = - 1

cot (x) sin (x) - cos^{2} (x) sec (x) = 0

\frac{cos ( 3 x )}{cos ( x )} = 4 cos^{2} (x) - 3

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