Question

If f(x) =sinx ,prove that 1/f(x)-f(x)=cotx.cosx

Answer 1

Answer:

Given:

f(x) = sinx

To prove;

1/f(x) - f(x) = cotx•cosx

Proof:

If f(x) = sinx , then 1/f(x) = 1/sinx.

Now;

LHS = 1/f(x) - f(x)

= 1/sinx - sinx

= {1 - (sinx)^2}/sinx

= (cosx)^2/sinx

{ 1-(sinx)^2 = (cosx)^2 }

= (cosx)(cosx/sinx)

= cosx•cotx

{ cosx/sinx = cotx }

= cotx•cosx

= RHS.

Hence proved.