Physics, Chemistry, Mathematics Forums

You may need to recheck the question,

We have,

\({\sin ^6}\theta+{\cos ^6}\theta= ({\sin^2}\theta+ {\cos ^2}\theta)({\sin^4}\theta- si{n^2}\theta co{s^2}\theta+ co{s^4}\theta)\)

\( = {\sin^4}\theta- si{n^2}\theta co{s^2}\theta+ co{s^4}\theta\)

\( = {({\sin ^2}\theta+ {\cos ^2}\theta)^2} - 3{\sin ^2}\theta{\cos ^2}\theta\)

\( = 1-3{\sin ^2}\theta{\cos ^2}\theta\) .............*

\(\sin \theta+ cos\theta= x\)

\( \Rightarrow 1 + 2\sin \theta\cos \theta= {x^2}\)
\( \Rightarrow \sin \theta cos\theta= {{{x^2} - 1} \over 2}\)

The above can be used in the equation marked * above to get the result.