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Transformation Formulae — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTransformation Formulae4 Marks
Question
$\text{If}\ \text{m}\sin\theta=\text{n}\sin(\theta+2\alpha),$ prove that $\tan(\theta+\alpha)\cot\alpha=\frac{\text{m+n}}{\text{m}-\text{n}}.$
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Answer

Given that $\text{m}\sin\theta=\text{n}\sin(\theta+2\alpha),$ We need to prove that $\tan(\theta+\alpha)=\frac{\text{m+n}}{\text{m}-\text{n}}\tan\alpha$ $\text{m}\sin\theta=\text{n}\sin(\theta+2\alpha)$ $\Rightarrow\ \frac{\sin(\theta+2\alpha)}{\sin\theta}=\frac{\text{m}}{\text{n}}$ Using Componendo - Dividendo, we have, $\Rightarrow\ \frac{\sin(\theta+2\alpha)+\sin\theta}{\sin(\theta+2\theta)-\sin\theta}=\frac{\text{m+n}}{\text{m}-\text{n}}...(1)$ We know that, $\sin\text{C}+\sin\text{D}=2\sin\frac{\text{C+D}}{2}\cos\frac{\text{C}-\text{D}}{2}$ and $\sin\text{C}-\sin\text{D}=2\cos\frac{\text{C+D}}{2}\sin\frac{\text{C}-\text{D}}{2}$ Applying the above formula in equation (1), we have, $\frac{2\sin\frac{\theta+2\theta+\theta}{2}\cos\frac{\theta+2\theta-\theta}{2}}{2\cos\frac{\theta+2\theta+\theta}{2}\sin\frac{\theta+2\theta-\theta}{2}}=\frac{\text{m+n}}{\text{m}-\text{n}}$ $\Rightarrow\ \frac{2\sin(\theta+\alpha)\cos\alpha}{2\cos(\theta+\alpha)\sin\alpha}=\frac{\text{m+n}}{\text{m}-\text{n}}$ $\Rightarrow\ \frac{\tan(\theta+\alpha)}{\tan\alpha}=\frac{\text{m+n}}{\text{m}-\text{n}}$ $\Rightarrow\ \tan(\theta+\alpha)=\frac{\text{m+n}}{\text{m}-\text{n}}\times\tan\alpha$ Hence proved.

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