Is (sin(θ)+cos(θ))^2=1+sin(2θ) ?

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

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prove (sin(θ)+cos(θ))^2=1+sin(2θ)

prove

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

True

Solution steps

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

Manipulating right 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

sin (x) + cos (x) = 2

sin (- \frac{3 π}{4})

4 cos^{2} (θ) - 3 = 0

tan (\frac{π}{6})

2 sin (2 x) = 3

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