Advertisements
Advertisements
Question
Solve for x : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .
Advertisements
Solution
\[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \tan^{- 1} 1\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x - 2}{x - 1} \right) = \tan^{- 1} 1 - \tan^{- 1} \left( \frac{x + 2}{x + 1} \right)\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x - 2}{x - 1} \right) = \tan^{- 1} \left( \frac{1 - \frac{x + 2}{x + 1}}{1 + \frac{x + 2}{x + 1}} \right)\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x - 2}{x - 1} \right) = \tan^{- 1} \left( \frac{x + 1 - x - 2}{x + 1 + x + 2} \right)\]
\[ \Rightarrow \tan^{- 1} \left( \frac{x - 2}{x - 1} \right) = \tan^{- 1} \left( \frac{- 1}{2x + 3} \right)\]
\[ \Rightarrow \frac{x - 2}{x - 1} = \frac{- 1}{2x + 3}\]
\[ \Rightarrow 2 x^2 + 3x - 4x - 6 = - x + 1\]
\[ \Rightarrow 2 x^2 = 1 + 6\]
\[ \Rightarrow x^2 = 7\]
\[ \Rightarrow x = \pm \sqrt{\frac{7}{2}}\]
APPEARS IN
RELATED QUESTIONS
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Prove `tan^(-1) 2/11 + tan^(-1) 7/24 = tan^(-1) 1/2`
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
Find the value of the given expression.
`tan(sin^(-1) 3/5 + cot^(-1) 3/2)`
Prove that:
`sin^(-1) 8/17 + sin^(-1) 3/5 = tan^(-1) 77/36`
Prove that:
`cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`
Solve the following equation for x: `cos (tan^(-1) x) = sin (cot^(-1) 3/4)`
Find: ∫ sin x · log cos x dx
If tan–1x + tan–1y + tan–1z = π, show that x + y + z = xyz
Prove that `tan^-1x + tan^-1 (2x)/(1 - x^2) = tan^-1 (3x - x^3)/(1 - 3x^2), |x| < 1/sqrt(3)`
Choose the correct alternative:
If `cot^-1(sqrt(sin alpha)) + tan^-1(sqrt(sin alpha))` = u, then cos 2u is equal to
Choose the correct alternative:
If `sin^-1x + cot^-1 (1/2) = pi/2`, then x is equal to
Evaluate tan (tan–1(– 4)).
Evaluate: `sin^-1 [cos(sin^-1 sqrt(3)/2)]`
Show that `2tan^-1 {tan alpha/2 * tan(pi/4 - beta/2)} = tan^-1 (sin alpha cos beta)/(cosalpha + sinbeta)`
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.
If cos–1x > sin–1x, then ______.
`"tan"^-1 1/3 + "tan"^-1 1/5 + "tan"^-1 1/7 = "tan"^-1 1/8 =` ____________.
If tan-1 2x + tan-1 3x = `pi/4,` then x is ____________.
If `"tan"^-1 (("x" - 1)/("x" + 2)) + "tan"^-1 (("x" + 1)/("x" + 2)) = pi/4,` then x is equal to ____________.
The value of `"tan"^-1 (1/2) + "tan"^-1(1/3) + "tan"^-1(7/8)` is ____________.
`"cos"^-1["cos"(2"cot"^-1(sqrt2 - 1))]` = ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
`"cos"^-1 (1/2)`
If `"sin"^-1 (1 - "x") - 2 "sin"^-1 ("x") = pi/2,` then x is equal to ____________.
If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x2 – 23x – 12 = 0
`tan^-1 sqrt3 - cot^-1 (- sqrt3)` is equal to ______.
If sin–1x + sin–1y + sin–1z = π, show that `x^2 - y^2 - z^2 + 2yzsqrt(1 - x^2) = 0`
Solve:
sin–1 (x) + sin–1 (1 – x) = cos–1 x
