Write the value of + 3A + 3A . - Vidyadip

Question

Write the value of $\frac{\sin\text{A}+\sin3\text{A}}{\cos\text{A}+\cos3\text{A}}.$

✓

Answer

We have,
$\frac{\sin\text{A}+\sin3\text{A}}{\cos\text{A}+\cos3\text{A}}=\frac{\sin3\text{A}+\sin\text{A}}{\cos3\text{A}+\cos\text{A}}$
$=\ \frac{2\sin\Big(\frac{3\text{A}+\text{A}}{2}\Big)\cos\Big(\frac{3\text{A}-\text{A}}{2}\Big)}{2\cos\Big(\frac{3\text{A}+\text{A}}{2}\Big)\cos\Big(\frac{3\text{A}-\text{A}}{2}\Big)}$
$=\ \frac{\sin2\text{A}\cos\text{A}}{\cos2\text{A}\cos\text{A}}$
$=\ \frac{\sin2\text{A}}{\cos2\text{A}}$
$=\ \tan2\text{A}$
$\therefore\ \frac{\sin\text{A}+\sin3\text{A}}{\cos\text{A}+\cos3\text{A}}=\tan2\text{A}.$

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