Question

If  \[\text{ sin } x + \text{ cos } x = a\], then find the value of

\[\sin^6 x + \cos^6 x\] .

 

 

Answer 1

Given: \[\text{ sin } x + \text{ cos } x = a\] Squaring on both sides, we get

\[\sin^2 x + \cos^2 x + 2\text{ sin } x\text { cos } x = a^2 \]
\[ \Rightarrow 1 + 2\text{ sin } x\text{ cos } x = a^2 \]

`⇒ sin x cos x = (a^2 - 1)/2`                .............(1)

Now,

\[\sin^6 x + \cos^6 x\]
\[ = \left( \sin^2 x + \cos^2 x \right)^3 - 3 \sin^2 x \cos^2 x\left( \sin^2 x + \cos^2 x \right)\]
\[ = 1 - 3 \left( \frac{a^2 - 1}{2} \right)^2 \left[ \text{ Using }  \left( 1 \right) \right]\]
\[ = \frac{4 - 3 \left( a^2 - 1 \right)^2}{4}\]

Hence, the required value is \[\frac{1}{4}\left[ 4 - 3 \left( a^2 - 1 \right)^2 \right]\] .