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33. If \( \tan x=\frac{\sin \alpha-\cos \alpha}{\sin \alpha+\cos \alpha} \), then show that \( \sin \alpha+\cos \alpha=\sqrt{2} \cos x \).[NCERT EXEMPLAR] |
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Answer» \(tan \,x = \frac{sin\alpha - cos\alpha}{sin\alpha + cos\alpha}\) \(= - \left(\frac{1- tan\,\alpha}{1+ tan\alpha}\right)\) \(= - tan(\frac\pi4 - \alpha)\) \(= tan(\alpha - \frac\pi4)\) \(\therefore x = \alpha - \frac\pi4\) \(\therefore cos \,x= cos\left(\alpha - \frac\pi4\right) = cos\left(\frac\pi4- \alpha\right)\) \(= cos\frac\pi4 cos\,\alpha + sin \frac\pi4 sin \,\alpha\) \(= \frac1{\sqrt2} (cos\, \alpha + sin\,\alpha)\) ⇒ \(\sqrt2 \, cos\,x = cos\,\alpha + sin\,\alpha\) Hence Proved. |
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