What is (cos(a)+cos(b))^2+(sin(a)-sin(b))^2 ?

The solution to (cos(a)+cos(b))^2+(sin(a)-sin(b))^2 is 2cos(a+b)+2

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simplify (cos(a)+cos(b))^2+(sin(a)-sin(b))^2

simplify

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

2 cos (a + b) + 2

Solution steps

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

Expand

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

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

Expand

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

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

Use the Pythagorean identity:

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

= 2 cos (a) cos (b) + 1 + 1 - 2 sin (a) sin (b)

Add the numbers:

1 + 1 = 2

= 2 cos (a) cos (b) + 2 - 2 sin (a) sin (b)

Use the Angle Sum identity:

cos (s) cos (t) - sin (s) sin (t) = cos (s + t)

= 2 (1 + cos (a + b))

Factor

2 cos (a) cos (b) + 2 - 2 sin (a) sin (b) : 2 (cos (a) cos (b) + 1 - sin (a) sin (b))

= 2 (cos (a + b) + 1)

Expand

2 (cos (a + b) + 1) : 2 cos (a + b) + 2

= 2 cos (a + b) + 2

\frac{d ^{2}}{d x ^{2}} ((y) (a x^{2} + b x + c x e^{- 3 x}))

\frac{( x - 2 ) ( x - 4 )}{3}

\frac{2 ( 6 - x ^{2} ) ^{2}}{x ^{2}}

2 π - 2 π x (2 - 2 x^{2})

\frac{2 - 2 x ^{2}}{( x ^{2} + 1 ) ^{2}}

What is (cos(a)+cos(b))^2+(sin(a)-sin(b))^2 ?
The solution to (cos(a)+cos(b))^2+(sin(a)-sin(b))^2 is 2cos(a+b)+2