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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

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

Apply Perfect Square Formula:

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

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

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

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

Simplify

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

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

Add similar elements:

cos^{2} (x) - cos^{2} (x) = 0

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

Add similar elements:

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

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

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

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

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

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

sin (t) cos (t) = 1

\frac{1}{1 - sin ( B )} = sec^{2} (B) + sec (B) tan (B)

tan (6 0^{\circ}) = \frac{2 tan ( 3 0 ^{\circ} )}{1 - tan ^{2} ( 3 0 ^{\circ} )}

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

cos (x) + sin (x) = 1

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