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

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

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

prove

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

True

Solution steps

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

Manipulating left side

1 + sin (2 x)

Use the Pythagorean identity:

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

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

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

Apply Perfect Square Formula:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (θ) - cos^{2} (θ) sin^{2} (θ) = sin^{4} (θ)

(1 + csc (θ)) (1 - csc (θ)) = - cot^{2} (θ)

cot (u) tan (u) - sin^{2} (u) = cos^{2} (u)

\frac{1}{cos ( θ )} = \frac{tan ( θ )}{sin ( θ )}

(tan (y) + cot (y)) sin (y) cos (y) = 1

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