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

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

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

simplify

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

- 2 cos (x - y)

Solution steps

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

Expand

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

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

Expand

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

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

Use the Pythagorean identity:

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

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

Add the numbers:

1 + 1 - 2 = 0

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

Use the Angle Difference identity:

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

= - 2 cos (x - y)

Factor out common term

2

= - 2 cos (x - y)

(8, - 5) y (- 3, 9)

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

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

(x - 6)^{4}

10 (y + 5 x) e^{y + 5 x} + e^{y + 5 x} \cdot 5 (y + 5 x)^{2}

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