Advertisements
Advertisements
Question
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
Solution
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))`
RELATED QUESTIONS
Find the principal solution of the following equation:
cosθ = `(1)/(2)`
Find the principal solution of the following equation:
Sec θ = `(2)/sqrt(3)`
Find the principal solution of the following equation:
cot θ = 0
Find the principal solution of the following equation:
sin θ = `-1/2`
Find the principal solution of the following equation:
tan θ = – 1
Find the general solution of the following equation:
sinθ = `1/2`.
Find the general solution of the following equation:
4 cos2 θ = 3
Find the general solution of the following equation:
sin θ = tan θ
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c2 .
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:
If cos pθ = cos qθ, p ≠ q, then ______.
Select the correct option from the given alternatives:
In ΔABC, ac cos B - bc cos A = _______
`"cos"^-1 ("cos" (7pi)/6)` = _________.
The principal value of sin–1 `(- sqrt3/2)` is ______.
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
`cos[tan^-1 1/3 + tan^-1 1/2]` = ______
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Find the principal solutions of the following equation:
sin 2θ = `-1/2`
Find the general solutions of the following equation:
`tan theta = - sqrt3`
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
Show that `tan^-1 1/2 - tan^-1 1/4 = tan^-1 2/9`.
Show that `cot^-1 1/3 - tan^-1 1/3 = cot^-1 3/4`.
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______
Find the principal solutions of cosec x = 2
Find the principal solutions of cos 2ЁЭСе = 1
If sin-1 x = `pi/10`, for some x ∈ [-1, 1], then the value of cos-1 x is _______.
`int (sin (log x))^2/x` log x dx = ?
If f(x) = sin-1`(sqrt((1 - x)/2))`, then f'(x) = ?
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
The general solution of the equation tan θ tan 2θ = 1 is given by ______
The general solution of cosec x = `-sqrt2` is ______
The general solution of x(1 + y2)1/2 dx + y(1 + x2)1/2 dy = 0 is ______.
Which of the following equation has no solution?
The number of principal solutions of tan 2θ = 1 is ______.
Principal solutions at the equation sin 2x + cos 2x = 0, where π < x < 2 π are ______.
The general solution to cos100x – sin100x = 1 is ______.
If `2sin^-1 3/7` = cos–1β, then find the value of β.
If tan θ + sec θ = `sqrt(3)`, find the general value of θ.
