What is (cos^3(x))(cos(x))-(sin^3(x))(sin(x)) ?

The solution to (cos^3(x))(cos(x))-(sin^3(x))(sin(x)) is cos(2x)

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simplify (cos^3(x))(cos(x))-(sin^3(x))(sin(x))

simplify

(cos^{3} (x)) (cos (x)) - (sin^{3} (x)) (sin (x))

cos (2 x)

Solution steps

(cos^{3} (x)) (cos (x)) - (sin^{3} (x)) (sin (x))

(cos^{3} (x)) (cos (x)) = cos^{4} (x)

sin^{3} (x) sin (x) = sin^{4} (x)

= cos^{4} (x) - sin^{4} (x)

Factor

cos^{4} (x) - sin^{4} (x) : (cos^{2} (x) + sin^{2} (x)) (cos (x) + sin (x)) (cos (x) - sin (x))

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

Rewrite using trig identities

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

Expand

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

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

Rewrite using trig identities

= cos (2 x)

(x^{4} + 3 x^{3} - 4 x^{2} + 5 x) (3 x^{4} - 8 x^{2})

(\frac{1}{2} \cdot x^{2} + \frac{1}{3} \cdot x - 1) (\frac{3}{4} \cdot x^{2} - \frac{2}{3} \cdot x + \frac{1}{9})

\frac{r ^{2} - r}{2 r ^{2} + 5 r - 7}

x^{3} - 3 x y (x - y) - y^{3} + 1

4 x^{2} - 10 2 x - 12

What is (cos^3(x))(cos(x))-(sin^3(x))(sin(x)) ?
The solution to (cos^3(x))(cos(x))-(sin^3(x))(sin(x)) is cos(2x)