Advertisements
Advertisements
प्रश्न
Show that `2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
Advertisements
उत्तर
`2 cot^(-1) 3/2 = 2 tan^(-1) 2/3 ...[because cot^(-1) x = tan^(-1) (1/x)]`
`= tan^(-1) [(2 × 2/3)/(1 - (2/3)^2)] ...[because 2 tan^(-1) x = tan^(-1) ((2x)/(1 - x^2))]`
`= tan^(-1) [(4/3)/(1 - 4/9)]`
`= tan^(-1) (4/3 × 9/5)`
`= tan^(-1) 12/5` ....(1)
Let `sec^(-1) 13/12 = α`
Then, `sec α = 13/12, "where" 0 < α < pi/2`
∴ `tan α > 0`
Now, `tan α = sqrt(sec^2 α - 1)`
`tan α = sqrt(169/144 - 1)`
`tan α = sqrt(25/144)`
`tan α = 5/12`
∴ `α = tan^(-1) 5/12 = cot^(-1) 12/5 .....[because tan^(-1) x = cot^(-1) (1/x)]`
∴ `sec^(-1) 13/12 = cot^(-1) 12/5` .....(2)
Now,
LHS = `2 cot^(-1) 3/2 + sec^(-1) 13/12`
LHS = `tan^(-1) 12/5 + cot^(-1) 12/5` ...[By (1) and (2)]
LHS = `π/2 .....[because tan^(-1) x + cot^(-1) x - π/2]`
RHS = `π/2`
LHS = RHS
`2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
Sec θ = `(2)/sqrt(3)`
Find the general solution of the following equation:
sinθ = `1/2`.
Find the general solution of the following equation:
4sin2θ = 1.
Find the general solution of the following equation:
cos 4θ = cos 2θ
Find the general solution of the following equation:
tan3θ = 3 tanθ.
State whether the following equation have solution or not?
3 tanθ = 5
In ΔABC, prove that `sin(("B" − "C")/2) = (("b" − "c")/"a")cos "A"/(2)`.
Select the correct option from the given alternatives:
The general solution of sec x = `sqrt(2)` is ______.
Select the correct option from the given alternatives:
In ΔABC if c2 + a2 – b2 = ac, then ∠B = ____.
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Select the correct option from the given alternatives:
If tan-1(2x) + tan-1(3x) = `pi/4`, then x = _____
Select the correct option from the given alternatives:
`2 "tan"^-1 (1/3) + "tan"^-1 (1/7) =` _____
Select the correct option from the given alternatives:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
The principal value branch of sec-1x is ______.
`cos[tan^-1 1/3 + tan^-1 1/2]` = ______
Select the correct option from the given alternatives:
In any ΔABC, if acos B = bcos A, then the triangle is
Find the principal solutions of the following equation:
sin 2θ = `-1/2`
Find the principal solutions of the following equation:
tan 3θ = - 1
Find the principal solutions of the following equation:
cot θ = 0
Find the general solutions of the following equation:
sin2 θ - cos2 θ = 1
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
Show that `tan^-1 1/2 = 1/3 tan^-1 11/2`
If | x | < 1, then prove that
`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`
If `tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2,` then show that xy + yz + zx = 1
Find the principal solutions of cos 2𝑥 = 1
Find the principal solutions of tan x = `-sqrt(3)`
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
`cos^-1 (cos (4pi)/3)` = ______.
`int (sin (log x))^2/x` log x dx = ?
The principal solutions of cot x = `sqrt3` are ______.
If 4 sin-1x + 6 cos-1 x = 3π then x = ______.
The measure of the angle between lines (sin2θ - 1)x2 - 2xy cos2θ + cos2θy2 = 0 is ______
If sin θ + cos θ = 1, then the general value of θ is ______.
The general solution of cot θ + tan θ = 2 is ______.
Find the principal solutions of cot θ = 0
If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
The general solution of the equation tan θ + tan 4θ + tan 7θ = tan θ tan 4θ tan 7θ is ______.
With usual notations, in any ΔABC, if a cos B = b cos A, then the triangle is ______.
If `tanx/(tan 2x) + (tan 2x)/tanx + 2` = 0, then the general value of x is ______.
