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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(cos (x) + sin (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)

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

We showed that the two sides could take the same form

\Rightarrow True

cos (θ) = - cos (- θ)

sin (θ) + cos (θ) = sin (θ + 4 5^{\circ})

\frac{sin ( 2 θ )}{sin ( θ )} = 2 cos (θ)

81 - 9 (3 sin (x))^{2} = 1

\frac{1 + sec ( x )}{csc ( x )} = sin (x) + tan (x)

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