# If sin x + "cosec" x = 2, then find sin^n x + "cosec" ^n x ?

Sep 3, 2016

${\sin}^{n} x + \frac{1}{{\sin}^{n} x} = 2$

#### Explanation:

Solving

$\sin x + \frac{1}{\sin} x = 2 \to {\sin}^{2} x - 2 \sin x + 1 = 0$

we have

$\sin x = 1$ then

${\sin}^{n} x + \frac{1}{{\sin}^{n} x} = 2$

This can be also demonstrated by finite induction.

1) First it is true for $n = 1$ (with sin x = 1 of course)
2) Supposing that it is true for $n$ then
3) Prove that it is true for $n + 1$

It is quite easy and it is left as an exercise.