Question

Differentiate sin−1(2x√1−x2) w.r.t. cos−1(1−x21+x2).

Answer 1

Answer:

(1 + x² ) / √(1 - x² )

Step-by-step explanation:

I think question is like this

Differentiate Sin⁻¹ (2x √1-x² ) w.r.t.

Cos⁻¹ ( 1 - x² / 1 + x² )

Solution--->

Let

y₁ = Sin⁻¹ { 2x √( 1 - x² ) }

Let x = Sinα => α = Sin⁻¹ x

y₁ = Sin⁻¹ { 2 Sinα √( 1 - Sin²α ) }

= Sin⁻¹ ( 2 Sinα √ Cos²α )

= Sin⁻¹ ( 2 Sinα Cosα )

= Sin⁻¹ ( Sin 2α )

= 2 α

Putting α = Sin⁻¹x we get

y₁ = 2 Sin⁻¹x

Differentiating with respect to x

dy₁ / dx = 2 d / dx ( Sin⁻¹x )

= 2 { 1 /√(1 - x²) }

Now

y₂ = Cos⁻¹ (1 - x² / 1 + x² )

Let x = tanβ => β = tan⁻¹ x

y₂ = Cos⁻¹ ( 1- tan²β / 1 + tan²β )

We have a formula

Cos2β = 1 - tan²β / 1 + tan²β

Applying it here we get

= Cos⁻¹ ( Cos 2β )

= 2β

Putting β = tan⁻¹x we get

y₂ = 2 tan⁻¹ x

Differentiating with respect to x

dy₂ / dx = 2 d / dx (tan⁻¹ x )

= 2 ( 1 / 1 + x² )

Now we have to differentiate y₁ with respect to y₂

dy₁ / dx

dy₁ / dy₂ = ---------------

dy₂ / dx

Putting value of dy₁/dx and dy₂/dx we get

2 / √( 1 - x² )

= -----------------------

2 / (1 + x² )

dy₁/dy₂= (1 + x² ) / √(1 - x² )