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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Simplify

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

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

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

Apply Perfect Square Formula:

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

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

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

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

Add similar elements:

2 cos (a) sin (a) - 2 cos (a) sin (a) = 0

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

cos (x) + sin (x) + tan (x) = sec (x)

81 - 9 (3 sin (x))^{2} = 4 sin (x) + 8

cot (x) \cdot cos (x) + sin (x) = csc (x)

(1 - cos^{2} (a)) \cdot (1 + tan^{2} (a)) = tan^{2} (a)

csc (x) \cdot tan (x) + sec (x) = 2 sec (x)

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