Question

Sin a+cosec a=2 then sin^2021 a-cosec^2021 a/sin^2021+cosec*2021a

Answer 1

SOLUTION

GIVEN

sin A + cosec A = 2

TO DETERMINE

The value of

$\displaystyle \sf \: \frac{ {\sin}^{2021} A - {cosec}^{2021} A}{{\sin}^{2021} A + {cosec}^{2021} A}$

EVALUATION

Here it is given that

$\displaystyle \sf \: \sin A + cosec A = 2$

$\displaystyle \sf \implies \sin A + \frac{1}{\sin A } = 2$

$\displaystyle \sf \implies \frac{ { \sin }^{2}A + 1 }{\sin A } = 2$

$\displaystyle \sf \implies { \sin }^{2}A + 1 - 2\sin A = 0$

$\displaystyle \sf \implies { (\sin A - 1)}^{2} = 0$

$\displaystyle \sf \implies { (\sin A - 1)}^{} = 0$

$\displaystyle \sf \implies \sin A = 1$

$\displaystyle \sf \implies cosec A = 1$

Now

$\displaystyle \sf \: \frac{ {\sin}^{2021} A - {cosec}^{2021} A}{{\sin}^{2021} A + {cosec}^{2021} A}$

$\displaystyle \sf \: = \frac{ {(1)}^{2021} - {(1)}^{2021} }{{(1)}^{2021} + {(1)}^{2021} }$

$\displaystyle \sf \: = \frac{1 - 1 }{1 + 1 }$

= 0

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