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Trigonometric Ratios Of Compounds — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios Of Compounds2 Marks
Question
Prove that: $1+\cos^22\text{x}=2(\cos^4\text{x}+\sin^4\text{x})$
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Answer

$\text{LHS}=1+\cos^22\text{x}$ $=1+(\cos^2\text{x}-\sin^2\text{x})$ $[\because\cos2\text{x}=\cos^2\text{x}-\sin^2\text{x}]$ $=1+\cos^4\text{x}+\sin^4\text{x}-2\sin^2\text{x}.\cos^2\text{x}$ $=(\sin^2\text{x}+\cos^2\text{x})^2+\cos^4\text{x}+\sin^4\text{x}-2\sin\text{x},\cos^2\text{x}$ $[\because\sin^2\text{x}+\cos^2\text{x}=1]$ $=\sin^4\text{x}+\cos^4\text{x}+2\sin^2\text{x}\cos^2\text{x}+\cos^4\text{x}+\sin^4\text{x}-2\sin^2\text{x}.\cos^2\text{x}$ $=2\big(\cos^4\text{x}+\sin^4\text{x}\big)\ \text{RHS}$

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