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Question Number 57448 by Pk1167156@gmail.com last updated on 04/Apr/19

If   sin x+cosec x=2, then sin^n x+cosec^n x  is equal to

$$\mathrm{If}\:\:\:\mathrm{sin}\:{x}+\mathrm{cosec}\:{x}=\mathrm{2},\:\mathrm{then}\:\mathrm{sin}^{{n}} {x}+\mathrm{cosec}^{{n}} {x} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Apr/19

a+(1/a)=2  a^2 +1=2a  (a−1)^2 =0  so a=1  sinx=a=1  cosecx=(1/a)=1  sin^n x+cosec^n x  (1)^n +(1)^n   =2

$${a}+\frac{\mathrm{1}}{{a}}=\mathrm{2} \\ $$$${a}^{\mathrm{2}} +\mathrm{1}=\mathrm{2}{a} \\ $$$$\left({a}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0}\:\:{so}\:{a}=\mathrm{1} \\ $$$${sinx}={a}=\mathrm{1} \\ $$$${cosecx}=\frac{\mathrm{1}}{{a}}=\mathrm{1} \\ $$$${sin}^{{n}} {x}+{cosec}^{{n}} {x} \\ $$$$\left(\mathrm{1}\right)^{{n}} +\left(\mathrm{1}\right)^{{n}} \\ $$$$=\mathrm{2} \\ $$

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