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

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

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

prove

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

True

Solution steps

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

Manipulating left 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)

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

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (2 x) = \frac{1}{2} (1 - cos (4 x))

\frac{sin ( θ ) tan ( θ )}{1 - cos ( θ )} = 1 + sec (θ)

sec (t) - \frac{cos ( t )}{1 + sin ( t )} = tan (t)

sec (x) - cos (x) = sin (x) \cdot tan (x)

sin (2 θ) - tan (θ) = tan (θ) cos (2 θ)

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