If + =a, find the value of ^6x+ ^6x. - Vidyadip

Question

If $\sin\text{x}+\cos\text{x}=\text{a},$ find the value of $\sin^6\text{x}+\cos^6\text{x}.$

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Answer

Given: $\sin\text{x}+\cos\text{x}=\text{a}$
squaring on both sides, we get
$\sin^2\text{x}+\cos^2\text{x}+2\sin\text{x}\cos\text{x}=\text{a}^2$
$\Rightarrow1+2\sin\text{x}\cos\text{x}=\text{a}^2$
$\Rightarrow\sin\text{x}\cos\text{x}=\frac{\text{a}^2-1}{2}\ .....(1)$
Now,
$\sin^6\text{x}+\cos^6\text{x}$
$=\big(\sin^2\text{x}+\cos^2\text{x}\big)^3-3\sin^2\text{x}\cos^2\text{x}(\sin^2\text{x}+\cos^2\text{x})$
$=1-3\big(\frac{\text{a}^2-1}{2}\big)^2$ [Using (1)]
$=\frac{4-3(\text{a}^2-2)^1}{4}$
Hence, the required value is $\frac{1}{4}\big[4-3(\text{a}^2-1)^2\big]$

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