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Trigonometric Ratios Of Compounds — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Ratios Of Compounds1 Mark
Question
If $\tan\frac{\text{x}}{2}=\frac{\text{m}}{\text{n}},$ then write the value of $\text{m}\ \sin\text{x}+\text{n}\cos\text{x}.$
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Answer

We have, $\tan\frac{\text{x}}{2}=\frac{\text{m}}{\text{n}}$ $\Rightarrow\frac{\sin\frac{\text{x}}{2}}{\cos\frac{\text{x}}{2}}=\frac{\text{m}}{\text{n}}$ $\Rightarrow\sin\frac{\text{x}}{2}=\text{mk},\&\cos\frac{\text{x}}{2}=\text{nk}(\text{say})$ Now, $\text{m}\sin\text{x}+\text{n}\cos\text{x}$ $=\text{m}2\sin\frac{\text{x}}{2},\cos\frac{\text{x}}{2}+\text{n}\Big(\cos^2\frac{\text{x}}{2}-\sin^2\frac{\text{x}}{2}\Big)$ $2\text{m}.\text{mk}.\text{nk}+\text{n}(\text{n}^2\text{k}^2-\text{m}^2\text{k}^2)$ $=2\text{m}^2\text{k}^2\text{n}+\text{nk}^2(\text{n}^2-\text{m}^2)$ $=\text{nk}^2(2\text{m}^2+\text{n}^2-\text{m}^2)$ $=\text{nk}^2(\text{m}^2+\text{n}^2)$ $=​​\text{n}(​\text{m}^2​\text{k}^2+​\text{n}^2​\text{k}^2)$ $​\text{n}=\Big(\sin^2\frac{​\text{x}}{2}+\cos^2\frac{​\text{x}}{2}\Big)$ $=​\text{n}$

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