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Trigonometric Functions — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Functions5 Marks
Question
 Prove the following:
$\cos^22\text{x}-\cos^26\text{x}=\sin8\text{x}\sin4\text{x}$
✓

Answer

It is known that $\cos\text{A}+\cos\text{B}=2\cos\Big(\frac{\text{A}+\text{B}}{2}\Big).\cos\Big(\frac{\text{A}-\text{B}}{2}\Big).$

$\cos\text{A}-\cos\text{B}=-2\sin\Big(\frac{\text{A}+\text{B}}{2}\Big).\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)$

$\therefore\text{L.H.S.}=\cos^22\text{x}-\cos^26\text{x}$

$=(\cos2\text{x}+\cos6\text{x})(\cos2\text{x}-6\text{x})$

$=\Big[2\cos\Big(\frac{2\text{x}+6\text{x}}{2}\Big)\cos\Big(\frac{2\text{x}-6\text{x}}{2}\Big)\Big]\Big[-2\sin\Big(\frac{2\text{x}+6\text{x}}{2}\Big)\sin\Big(\frac{2\text{x}-6\text{x}}{2}\Big)\Big]$

$=[2\cos4\text{x}\cos(-2\text{x})][-2\sin4\text{x}\sin(-2\text{x})]$

$=[2\cos4\text{x}\cos2\text{x}][-2\sin4\text{x})(\sin-2\text{x})]$

$=(2\cos4\text{x}\cos4\text{x})(2\sin2\text{x}\cos2\text{x})$

$=\sin8\text{x}\sin4\text{x}$

$=\text{R.H.S.}$

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