Question

Prove that:(its urgent plz help someone who's intelligent enough !!)
$cos. sin^-^1.cot. cosec^-^1x= \sqrt{2- x^{2} }$

Answer 1

To prove, Cos [ Sin⁻¹ { Cot (Cosec⁻¹ x) } ] = √(2 - x²)

Let x = Cosec A
we know that  Cosec² A - Cot² A = 1.
  Hence,  Cot A = √(x² - 1)

Thus LHS = Cos [ Sin⁻¹ √(x² - 1) ]
 
  Let  Sin⁻¹ √(x² - 1) = y
        Sin y = √(x² - 1)
        Cos² y = 1 - Sin² y  =  1 - (x² - 1) = 2 - x²

 Hence Cos y = √(2 - x²)   = RHS