What is (cos(x)+cos(y))^2+(sin(x)-sin(y))^2 ?

The solution to (cos(x)+cos(y))^2+(sin(x)-sin(y))^2 is 2cos(x+y)+2

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simplify (cos(x)+cos(y))^2+(sin(x)-sin(y))^2

simplify

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

2 cos (x + y) + 2

Solution steps

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

Expand

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

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

Expand

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

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

Use the Pythagorean identity:

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

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

Add the numbers:

1 + 1 = 2

= 2 cos (x) cos (y) - 2 sin (x) sin (y) + 2

Use the Angle Sum identity:

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

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

Factor

2 cos (x) cos (y) - 2 sin (x) sin (y) + 2 : 2 (cos (x) cos (y) - sin (x) sin (y) + 1)

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

Expand

2 (cos (x + y) + 1) : 2 cos (x + y) + 2

= 2 cos (x + y) + 2

\frac{2 x ^{3} - 4 x ^{2} - 15 x + 5}{( x - 1 ) ^{2}}

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

(5 + 3 z i)^{2}

\frac{3 x ^{3} + 10 x ^{2} + 27 x + 27}{x ^{2} ( x + 3 ) ^{2}}

- x^{\frac{2}{3}} (x - 5)

What is (cos(x)+cos(y))^2+(sin(x)-sin(y))^2 ?
The solution to (cos(x)+cos(y))^2+(sin(x)-sin(y))^2 is 2cos(x+y)+2