Question
Prove that:
$\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$
$\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$
✓
Answer
We have,
$\text{LHS}=\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin2\text{A}+\sin2\text{A}}$
$=\ \frac{(\cos4\text{A}+\cos2\text{A})+\cos3\text{A}}{(\sin4\text{A}+\sin2\text{A})+\sin3\text{A}}$
$=\ \frac{2\cos\Big(\frac{4\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{4\text{A}-2\text{A}}{2}\Big)+\cos3\text{A}}{2\sin\Big(\frac{4\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{4\text{A}-2\text{A}}{2}\Big)+\sin3\text{A}}$
$=\ \frac{2\cos3\text{A}\cos\text{A}+\cos3\text{A}}{2\sin3\text{A}\cos\text{A}+\sin3\text{A}}$
$=\ \frac{\cos3\text{A}(2\cos\text{A}+1)}{\sin3\text{A}(2\cos2\text{A}+1)}$
$=\ \frac{\cos3\text{A}}{\sin\text{A}}$
$=\ \cot3\text{A}$
$=\ \text{RHS}$
$\therefore\ \frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$ Hence proved.
$\text{LHS}=\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin2\text{A}+\sin2\text{A}}$
$=\ \frac{(\cos4\text{A}+\cos2\text{A})+\cos3\text{A}}{(\sin4\text{A}+\sin2\text{A})+\sin3\text{A}}$
$=\ \frac{2\cos\Big(\frac{4\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{4\text{A}-2\text{A}}{2}\Big)+\cos3\text{A}}{2\sin\Big(\frac{4\text{A}+2\text{A}}{2}\Big)\cos\Big(\frac{4\text{A}-2\text{A}}{2}\Big)+\sin3\text{A}}$
$=\ \frac{2\cos3\text{A}\cos\text{A}+\cos3\text{A}}{2\sin3\text{A}\cos\text{A}+\sin3\text{A}}$
$=\ \frac{\cos3\text{A}(2\cos\text{A}+1)}{\sin3\text{A}(2\cos2\text{A}+1)}$
$=\ \frac{\cos3\text{A}}{\sin\text{A}}$
$=\ \cot3\text{A}$
$=\ \text{RHS}$
$\therefore\ \frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$ Hence proved.
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