Question

The value of cot (tan^-1 2x + cot^-1 2x) is

Answer 1

The value of cot(tan¯¹ 2x + cot¯¹ 2x ) is..

solution : let cot¯¹2x = A ⇒cotA = 2x

⇒tanA = 1/2x

⇒tan¯¹(1/2x) = A

so, cot¯¹2x = tan¯¹(1/2x)

now tan¯¹ 2x + cot¯¹ 2x = tan¯¹ 2x + tan¯¹ (1/2x)

we know, tan¯¹x + tan¯¹y = tan¯¹(x + y)/(1 - xy)

so, tan¯¹ 2x + tan¯¹ (1/2x) = tan¯¹ [(2x + 1/2x)/(1 - 2x × 1/2x)] = tan¯¹ (∞) = π/2

so, tan¯¹ 2x + cot¯¹ 2x = π/2

now cot(tan¯¹ 2x + cot¯¹ 2x) = cot(π/2) = 0

Therefore the value of cot(tan¯¹ 2x + cot¯¹ 2x) is 0