If sin x + cos x = a, Find the Value of sin⁶x + cos⁶x
Question
If
\[ \sin x+\cos x=a, \]find the value of
\[ \sin^6x+\cos^6x. \]Solution
Given
\[ \sin x+\cos x=a. \]Squaring both sides,
\[ a^2 = \sin^2x+\cos^2x+2\sin x\cos x \] \[ a^2 = 1+2\sin x\cos x \] \[ \sin x\cos x = \frac{a^2-1}{2}. \]Now use the identity
\[ u^3+v^3=(u+v)^3-3uv(u+v), \]where
\[ u=\sin^2x,\qquad v=\cos^2x. \]Then
\[ \sin^6x+\cos^6x = (\sin^2x+\cos^2x)^3 -3\sin^2x\cos^2x(\sin^2x+\cos^2x) \] \[ = 1-3\sin^2x\cos^2x. \]Substituting
\[ \sin^2x\cos^2x = \left(\frac{a^2-1}{2}\right)^2, \]we get
\[ \sin^6x+\cos^6x = 1-\frac{3}{4}(a^2-1)^2. \]