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Trigonometric Ratios Of Compound Angles — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Ratios Of Compound Angles2 Marks
Question
Prove that:
$1+\cos^22\text{x}=2(\cos^4\text{x}+\sin^4\text{x})$
✓

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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