Advertisements
Advertisements
प्रश्न
If tan-1 x - cot-1 x = tan-1 `(1/sqrt(3)),`x> 0 then find the value of x and hence find the value of sec-1 `(2/x)`.
Advertisements
उत्तर १
tan-1 x - cot-1 x = tan-1 `(1/sqrt(3)),`x> 0
⇒ tan-1 x - tan-1 `(1/"x")` = tan-1 `(1/sqrt(3))` ....[∵ cot-1 "x" = tan-1 `(1/"x"), "x" >0`]
⇒`tan^-1 (("x"-1/"x")/(1+"x". 1/"x")) = tan^-1 (1/sqrt3)`
⇒ `("x"^2 - 1)/(2"x") = 1/sqrt(3)`
⇒ `sqrt3"x"^2 - 2"x" - sqrt(3) = 0`
⇒ `sqrt3"x"^2 - 3"x" + "x" -sqrt(3) = 0`
⇒ `sqrt3x ("x" -sqrt3) + 1 ("x" - sqrt3) = 0`
⇒`(x - sqrt3) (sqrt3"x" + 1 ) =0`
⇒ `"x" = - 1/sqrt3, sqrt3`
∵ x >0, x = `sqrt3`
⇒ `sec^-1 (2/"x") = sec^-1 (2/sqrt3)`
⇒ `sec^-1 (2/"x") = sec^-1 (sec π/(6))`
⇒ `sec^-1 (2/"x") = π/6`
उत्तर २
Given,
tan-1 x - cot-1 x = tan-1 `(1/sqrt3),` x > 0
⇒ `tan^-1 x - tan^-1 (1/x) = tan^-1 (1/sqrt3) ....[ ∵ cot^-1 x = tan-1 (1/x), x > 0 ] `
⇒`tan^-1 ((x-1/x)/(1+x. 1/x)) = tan^-1 (1/sqrt3)`
⇒ `("x"^2 - 1)/(2"x") = 1/sqrt(3)`
⇒ `sqrt3"x"^2 - 2"x" - sqrt(3) = 0`
⇒ `sqrt3"x"^2 - 3"x" + "x" -sqrt(3) = 0`
⇒ `sqrt3x ("x" -sqrt3) + 1 ("x" - sqrt3) = 0`
⇒`(x - sqrt3) (sqrt3"x" + 1 ) =0`
⇒ `"x" = - 1/sqrt3, sqrt3`
∵ x >0, x = `sqrt3`
⇒ `sec^-1 (2/"x") = sec^-1 (2/sqrt3)`
⇒ `sec^-1 (2/"x") = sec^-1 (sec π/(6))`
⇒ `sec^-1 (2/"x") = π/6`
APPEARS IN
संबंधित प्रश्न
Prove that `2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)=pi/4`
Write the function in the simplest form: `tan^(-1) 1/(sqrt(x^2 - 1)), |x| > 1`
Find the value of the given expression.
`tan(sin^(-1) 3/5 + cot^(-1) 3/2)`
Prove that:
`tan^(-1) 63/16 = sin^(-1) 5/13 + cos^(-1) 3/5`
Prove that:
`cot^(-1) ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx))) = x/2, x in (0, pi/4)`
If cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.
Find the value, if it exists. If not, give the reason for non-existence
`tan^-1(sin(- (5pi)/2))`
Prove that `tan^-1 2/11 + tan^-1 7/24 = tan^-1 1/2`
If tan–1x + tan–1y + tan–1z = π, show that x + y + z = xyz
Simplify: `tan^-1 x/y - tan^-1 (x - y)/(x + y)`
Solve: `sin^-1 5/x + sin^-1 12/x = pi/2`
Evaluate: `tan^-1 sqrt(3) - sec^-1(-2)`.
Prove that `2sin^-1 3/5 - tan^-1 17/31 = pi/4`
If `sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2))`. where a, x ∈ ] 0, 1, then the value of x is ______.
If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.
The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.
The minimum value of sinx - cosx is ____________.
The value of `"tan"^-1 (1/2) + "tan"^-1 (1/3) + "tan"^-1 (7/8)` is ____________.
`"cot" ("cosec"^-1 5/3 + "tan"^-1 2/3) =` ____________.
If tan-1 2x + tan-1 3x = `pi/4,` then x is ____________.
The value of `"tan"^-1 (1/2) + "tan"^-1(1/3) + "tan"^-1(7/8)` is ____________.
The value of `"tan"^-1 (3/4) + "tan"^-1 (1/7)` is ____________.
`"tan" (pi/4 + 1/2 "cos"^-1 "x") + "tan" (pi/4 - 1/2 "cos"^-1 "x") =` ____________.
If `6"sin"^-1 ("x"^2 - 6"x" + 8.5) = pi,` then x is equal to ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
`"sin"^-1 ((-1)/2)`
The Simplest form of `cot^-1 (1/sqrt(x^2 - 1))`, |x| > 1 is
Find the value of `sin^-1 [sin((13π)/7)]`
The set of all values of k for which (tan–1 x)3 + (cot–1 x)3 = kπ3, x ∈ R, is the internal ______.
The value of cosec `[sin^-1((-1)/2)] - sec[cos^-1((-1)/2)]` is equal to ______.
