Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios Of Compounds1 Mark
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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