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

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

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

prove

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

True

Solution steps

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

Manipulating right side

1 + sin (2 t)

Use the Pythagorean identity:

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

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

Use the Double Angle identity:

sin (2 t) = 2 sin (t) cos (t)

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

Apply Perfect Square Formula:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (θ) = \frac{1}{cos ^{2} ( θ )}

csc^{2} (θ) = \frac{1}{sin ^{2} ( θ )}

sin (x) = \frac{1}{2}

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

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

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