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Question Number 216421 by MATHEMATICSAM last updated on 07/Feb/25

$$\mathrm{If}a\sin \theta +b\cos \theta =a\mathrm{cosec}\theta +b\sec \theta \mathrm{then}$$$$\mathrm{prove}\mathrm{that}\mathrm{each}\mathrm{term}\mathrm{is}\mathrm{equal}\mathrm{to}$$$$(a^{\frac{2}{3}}-b^{\frac{2}{3}})\sqrt{a^{\frac{2}{3}}+b^{\frac{2}{3}}}.$$

Answered by mr W last updated on 07/Feb/25

$$a\sin \theta +b\cos \theta =\frac{a}{\sin \theta }+\frac{b}{\cos \theta }=k,say$$$$a\sin \theta +b\cos \theta =k...\left(i\right)$$$$a\cos \theta +b\sin \theta =k\sin \theta \cos \theta ...\left(ii\right)$$$$a\cos \theta +b\sin \theta =(a\sin \theta +b\cos \theta )\sin \theta \cos \theta$$$$a\cos \theta (1-{\sin }^2\theta )+b\sin \theta (1-{\cos }^2\theta )=0$$$$a{\cos }^3\theta +b{\sin }^3\theta =0$$$$\Rightarrow \tan \theta =-\frac{a^{\frac{1}{3}}}{b^{\frac{1}{3}}}$$$$\Rightarrow \sin \theta =\pm \frac{a^{\frac{1}{3}}}{\sqrt{a^{\frac{2}{3}}+b^{\frac{2}{3}}}}$$$$\Rightarrow \cos \theta =\mp \frac{b^{\frac{1}{3}}}{\sqrt{a^{\frac{2}{3}}+b^{\frac{2}{3}}}}$$$$k=\frac{a}{\sin \theta }+\frac{b}{\cos \theta }$$$$=\pm (a^{\frac{2}{3}}-b^{\frac{2}{3}})\sqrt{a^{\frac{2}{3}}+b^{\frac{2}{3}}}✓$$

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