Maths · Inverse trigonometrical functions and their properties
If +\cot ^{-1}\left(\frac{1}{x-1}\right)= \) then
If \( \cot ^{-1}\left(\frac{1}{x+1}\right)+\cot ^{-1}\left(\frac{1}{x-1}\right)= \) \( \tan ^{-1} 3 x-\tan ^{-1} x \) then \( \boldsymbol{x}= \)
- A. \( \pm 1 / 2 \)
- B. \( -1, \pm 1 / 3 \)
- C. 2,±1
- D. \( -1 . \pm 1 / 2 \)
Step-by-step solution
Transform cot^{-1}(1/(x+1)) and cot^{-1}(1/(x-1)) into tan^{-1}(x+1) and tan^{-1}(x-1) using the identity cot^{-1}(1/a)=tan^{-1}(a) for a>0, with appropriate adjustments for sign. Simplify the equation using the formula for tan^{-1}(u) - tan^{-1}(v). Taking tangent on both sides yields 2x/(2-x^2)=2x/(1+3x^2). Solving gives x=0 or x=±1/2. x=0 does not satisfy the original equation, so x=±1/2 are the solutions, though domain conditions must be considered.
Related MCQs
- Assertion Consider =\sin ^{-1}\left(\sec \left(\tan ^{-1} \boldsymbol{x}\right)+\right. \) \right. \) Statement-1: Domain of \) is a singlet…
- If value of which satisfy equation ^{2}-3\left(\cot ^{-1} x\right)+2>0 \) is or Find the value of…
- The number of real solutions of the equation }+ \) is…
- Assertion \) If then +\cos ^{-1}(\sin x)=\pi-2 x \) Reason \cos ^{-1} x=\frac{\pi}{2}-\sin ^{-1} x \forall x \in \) \)…
- Assertion then Reason \)…