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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(1 + sin (y)) (1 + sin (- y))

Use the negative angle identity:

sin (- x) = - sin (x)

= (1 - sin (y)) (1 + sin (y))

Rewrite using trig identities

(1 - sin (y)) (1 + sin (y))

Expand

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

(1 + sin (y)) (1 - sin (y))

Apply Difference of Two Squares Formula:

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

a = 1, b = sin (y)

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (y)

= 1 - sin^{2} (y)

Use the Pythagorean identity:

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

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

= cos^{2} (y)

= cos^{2} (y)

We showed that the two sides could take the same form

\Rightarrow True

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