Question

prove that cot 4x (sin 5x + sin 3x )= cot x ( sin 5x - sin 3x)​

Answer 1

Answer:

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Answer 2

AnswEr:

LHS = cot 4x (sin 5x + sin 3x)

$\\ \implies \sf \: cos \: 4x \times 2 \: sin( \frac{5x + 3x}{2} ) \: cos \: ( \frac{5x - 3x}{2} ) \\ \\ \\ \implies \sf \: \frac{cos \: 4x}{sin \: 4x} \times 2 \: sin \: x \: cos \: 4x \\ \\ \\ \implies \sf \: 2 \: cos \: 4x \: cos \: x \qquad \qquad \: (i)$

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RHS = cot x (sin 5x - sin 3x)

$\\ \sf \implies \: cot \: x \times 2 \: sin( \frac{5x - 3x}{2} ) \: cos \: ( \frac{5x + 3x}{2} ) \\ \\ \\ \implies \sf \frac{cos \: x}{sin \: x} \times 2 \: sin \: x \: cos \: 4x \\ \\ \\ \implies \sf \: 2 \: cos \: 4x \: cos \: x \qquad \qquad \: (2)$

From (1) and (2), we obtain LHS = RHS