Question

Simplify cot^-1 ( 1 / √(x2−1) ) for x < -1.

Answer 1

Answer:

Sec⁻¹ x

Step-by-step explanation:

Given-----> Cot⁻¹ { 1 / √( x² - 1 ) }

To find------> Simplify given function

Solution-----> Let,

y = Cot⁻¹ { 1 / √( x² - 1 ) }

=> y = tan⁻¹ {√( x² - 1 ) }

Putting , x = Secθ

=> Sec⁻¹ ( x ) = Sec⁻¹ ( Secθ )

=> Sec⁻¹ ( x ) = θ

Now , y = tan⁻¹ [ √{ ( Secθ )² - 1 ]

=> y = tan⁻¹ { √( Sec²θ - 1 ) }

We know that , tan² A = Sec²A - 1 , applying it we get,

=> y = tan⁻¹ { √ tan²θ }

=> y = tan⁻¹ ( tanθ )

=> y = θ

Putting θ = Sec⁻¹ x , in it we get ,

=> y = Sec⁻¹ x

Additional information----->

1) Sin⁻¹ x + Cos⁻¹ x = π/2

2) tan⁻¹ x + Cot¹ x = π/2

3) Sec⁻¹ x + Cosec⁻¹ x = π/2

4) tan⁻¹ x + tan⁻¹ y = tan⁻¹ ( x+ y ) / ( 1 - xy )

5) tan⁻¹ x - tan⁻¹ y = tan⁻¹ ( x - y ) / ( 1 + xy )