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

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

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

prove

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

True

Solution steps

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

Manipulating right side

sin (2 x) + 1

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

\frac{tan ( θ )}{sec ( θ )} = sin (θ)

\frac{tan ( x )}{sin ( x )} = sec (x)

(1 - sin^{2} (θ)) sec^{2} (θ) = 1

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

sin^{3} (θ) = \frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

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