Is 1+sin(2θ)=(sin(θ)+cos(θ))^2 ?

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

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prove 1+sin(2θ)=(sin(θ)+cos(θ))^2

prove

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

True

Solution steps

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

Manipulating left side

1 + sin (2 θ)

Use the Pythagorean identity:

1 = cos^{2} (θ) + sin^{2} (θ)

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

Use the Double Angle identity:

sin (2 θ) = 2 sin (θ) cos (θ)

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

Apply Perfect Square Formula:

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

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

= (sin (θ) + cos (θ))^{2}

We showed that the two sides could take the same form

\Rightarrow True

cos (\frac{5 π}{4} - x) = - \frac{2}{2} (cos (x) + sin (x))

\frac{sin ^{2} ( x )}{cos ( x )} = sec (x) - cos (x)

5 cos^{2} (θ) + 3 sin^{2} (θ) = 3 + 2 cos^{2} (θ)

cos (x + \frac{π}{3}) + cos (x - \frac{π}{3}) = cos (x)

3 cos (x + y) + 3 cos (x - y) = 6 cos (x) cos (y)

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