Question

If sin ( cot-1 (x + 1) ) = cos ( tan-1x ) ...... find x

Answer 1

We have to find the value of x if sin(cot¯¹(x + 1)) = cos(tan¯¹x)

Solution : here sin(cot¯¹(x + 1)) = cos(tan¯¹x)

we know, sin(π/2 - θ) = cosθ

so, cos(tan¯¹x) = sin(π/2 - tan¯¹x)

Now, sin(cot¯¹(x + 1)) = cos(tan¯¹x) = sin(π/2 - tan¯¹x)

⇒cot¯¹(x + 1) = π/2 - tan¯¹x

⇒cot¯¹(x + 1) + tan¯¹x = π/2

Case 1 : Let's take x > 0

cot¯¹(x + 1) = tan¯¹1/(x + 1)

⇒tan¯¹1/(x + 1) + tan¯¹x = π/2

⇒tan¯¹[{1/(x + 1) + x}/{1 - x/(x + 1)}] = π/2

⇒{(1 + x² + x)}/1 = tanπ/2 = 1/0

0 = 1 , not possible solution.

Case 2 : let's take x < 0

Then, cot¯¹(x + 1) = -tan¯¹1/(x + 1)

so, -tan¯¹1/(x + 1) + tan¯¹ x = π/2

⇒tan¯¹[{x - 1/(x + 1) }/{1 + x/(x + 1)}] = π/2

⇒(x² + x - 1)/(x + 1 + x) = tanπ/2 = 1/0

⇒2x + 1 = 0

⇒x = -1/2

therefore the value of x is -1/2