Question

If sin x + cosec x = 2, then write the value of sinⁿ x + cosecⁿ x.

 

Answer 1

We have: 
$$\sin x+cosecx=2$$
$$\Rightarrow \sin x+\frac{1}{\sin x}=2$$
$$\Rightarrow \frac{{\sin }^{2}x+1}{\sin x}=2$$
${\displaystyle \Rightarrow {\sin }^{2}x+1=2\sin x}$

${\displaystyle \Rightarrow {\sin }^{2}x+1-2\sin x=0}$

${\displaystyle \Rightarrow {(\sin x-1)}^2=0}$

${\displaystyle \Rightarrow \sin x-1=0}$

${\displaystyle \Rightarrow \sin x=1}$

And ${\displaystyle \cos ecx=\frac{1}{\sin x}=1}$

${\displaystyle \therefore {\sin }^{n}x+\cos e{c}^{n}x=1^n+1^n}$

= 1 + 1 = 2