Advertisements
Advertisements
प्रश्न
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))`
Advertisements
उत्तर
Let tan-1x = y
Then, x = tan y
Now, `tan^-1 ("2x"/(1 - "x"^2)) = tan^-1 (("2 tan y")/(1 - tan^2 "y"))`
`= tan^-1 (tan 2"y")`
= 2y
= 2 tan-1x ......(1)
`sin^-1 ("2x"/(1 + "x"^2)) = sin^-1 (("2 tan y")/(1 + tan^2 "y"))`
`= sin^-1 (sin 2"y")`
= 2y
= 2 tan-1x ......(2)
`cos^-1 ((1 - "x"^2)/(1 + "x"^2)) = cos^-1 ((1 - tan^2 "y")/(1 + tan^2 "y"))`
`= cos^-1 (cos "2y")`
= 2y
`= 2 tan^-1 "x"` ......(3)
From (1), (2) and (3), we get
`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`
संबंधित प्रश्न
Find the principal solution of the following equation:
cot θ = 0
Find the general solution of the following equation:
tan θ = `(1)/(sqrt(3))`
Find the general solution of the following equation:
sin 2θ = `1/2`
Find the general solution of the following equation:
4 cos2 θ = 3
Find the general solution of the following equation:
cos 4θ = cos 2θ
Find the general solution of the following equation:
sin θ = tan θ
State whether the following equation have solution or not?
3 tanθ = 5
In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.
In ΔABC, if a cos A = b cos B then prove that the triangle is either a right angled or an isosceles traingle.
Select the correct option from the given alternatives:
If polar coordinates of a point are `(2, pi/4)`, then its cartesian coordinates are
Select the correct option from the given alternatives:
In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are
Select the correct option from the given alternatives:
In ΔABC if c2 + a2 – b2 = ac, then ∠B = ____.
`"cos"^-1 ("cos" (7pi)/6)` = _________.
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Find the principal solutions of the following equation:
tan 3θ = - 1
Find the general solutions of the following equation:
sin θ - cos θ = 1
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
With the usual notations, prove that `(sin("A" - "B"))/(sin ("A" + "B")) = ("a"^2 - "b"^2)/"c"^2`
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
Show that `cot^-1 1/3 - tan^-1 1/3 = cot^-1 3/4`.
Show that `tan^-1 1/2 = 1/3 tan^-1 11/2`
Show that `2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
If x, y, z are positive, then prove that
`tan^-1 (("x - y")/(1 + "xy")) + tan^-1 (("y - z")/(1 + "yz")) + tan^-1 (("z - x")/(1 + "zx")) = 0`
If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.
`int 1/(sin x * cos x)` dx = ?
`int (sin (log x))^2/x` log x dx = ?
If tan-1 x + 2cot-1 x = `(5pi)/6`, then x is
If f(x) = sin-1`(sqrt((1 - x)/2))`, then f'(x) = ?
If 2 cos2 θ + 3 cos θ = 2, then permissible value of cos θ is ________.
The number of solutions of `sin^2 theta = 1/2` in [0, π] is ______.
Which of the following equations has no solution?
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
If function
f(x) = `x - |x|/x, x < 0`
= `x + |x|/x, x > 0`
= 1, x = 0, then
If 4 sin-1x + 6 cos-1 x = 3π then x = ______.
The general solution of sin 2x = cos 2x is ______
The general solution of 4sin2 x = 1 is ______.
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied, if ______.
The number of principal solutions of tan 2θ = 1 is ______.
The general solution to cos100x – sin100x = 1 is ______.
If `2sin^-1 3/7` = cos–1β, then find the value of β.
