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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Expand

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

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

Apply Perfect Square Formula:

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

a = cos (x), b = sin (x)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin (2 x) = 2 cos (x)

sec (β) sin (β) cot (β) = 1

1 - sin^{2} (θ) + sin^{4} (θ) = cos^{4} (θ)

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

3 - 6 sin^{2} (x) = - 3 cos (2 x)

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