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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(cos (θ) + sin (θ))^{2}

Expand

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

(cos (θ) + sin (θ))^{2}

Apply Perfect Square Formula:

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

a = cos (θ), b = sin (θ)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= 1 + 2 cos (θ) sin (θ)

= 1 + 2 cos (θ) sin (θ)

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (θ) = \frac{3}{4} sin (θ) - \frac{1}{4} sin (3 θ)

\frac{1 + tan ^{2} ( x )}{cos ^{2} ( x )} = sec^{4} (x)

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

sin^{3} (x) + cos^{3} (x) tan (x) = sin (x)

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

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