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Trigonometric Functions — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions1 Mark
MCQ
The value of $\frac{\sin5\alpha-\sin\beta}{\cos5\alpha+2\cos4\alpha+\cos3\alpha}$ is:
  • A
    $\cot\frac{\alpha}{2}$
  • B
    $\cot\alpha$
  • ✓
    $\tan\frac{\alpha}{2}$
  • D
    None of these
✓

Answer

Correct option: C.
$\tan\frac{\alpha}{2}$
$\frac{\sin5\alpha-\sin3\alpha}{\cos5\alpha+2\cos4\alpha+\cos3\alpha}$
$=\frac{\sin5\alpha-\sin3\alpha}{\cos5\alpha+\cos3\alpha+2\cos4\alpha}$
$=\frac{2\sin\alpha\cos4\alpha}{2\cos4\alpha\cos\alpha+2\cos4\alpha}$
$=\frac{2\sin\alpha\cos4\alpha}{2\cos4\alpha(\cos\alpha+1)}$
$=\frac{\sin\alpha}{\cos\alpha+1}$
$=\frac{2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}{\cos^2\frac{\alpha}{2}-\sin^2\frac{\alpha}{2}+\sin^2\frac{\alpha}{2}+\cos^2\frac{\alpha}{2}}$
$=\frac{2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}{2\cos^2\frac{\alpha}{2}}$
$=\frac{\sin\frac{\alpha}{2}}{\cos^2\frac{\alpha}{2}}$
$=\tan\frac{\alpha}{2}$

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