Question

Cot inverse 2x + cot inverse 3x = pie /4

Answer 1

your answer is
$x = 1$

Answer 2

Given:

Cot inverse 2x + cot inverse 3x = pie /4

To Find:

The value of x .

Solution:

• $cot^{-1}$ 2x +  $cot^{-1}$3x = pie /4
• $tan^{-1}$(1/2x) + $tan^{-1}$(1/3x) = π/4

We know that , for any angle a and b ,

• $tan^{-1}$a + $tan^{-1}$b = $tan^{-1} \frac{a+b}{1- ab}$

Now we apply this in the above equation,

• $tan^{-1}$ ( 1/2x + 1/3x) / ( 1 - 1/6x²)
• π/4 = $tan^{-1}$ ( 5x/6x² / ( 6x² - 1)/6x²)
• tan π/4 = 5x/6x² -1
• 1 = 5x/6x² -1

Therefore,

• 6x² - 1 = 5x
• 6x² -5x -1 = 0

We need to solve this quadratic equation to find the value of x .

By splitting the middle term-5 into -6 + 1 ,

• 6x² - 6x + x - 1 = 0
• 6x( x -1 ) +1 ( x -1) = 0
• (6x + 1)(x -1) = 0
• x = -1/6 and x = 1 are the solutions.
• x  cannot be -1/6 since $tan^{-1}$( negative value)  = - $tan^{-1}$ ( value)

Value of x for which Cot inverse 2x + cot inverse 3x = pie /4 , is x =1