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:
Sec θ = `(2)/sqrt(3)`
Find the general solution of the following equation:
tan θ = - 1
Find the general solution of the following equation:
sin 2θ = `1/2`
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
sin θ = tan θ
Find the general solution of the following equation:
tan3θ = 3 tanθ.
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation have solution or not?
3 tanθ = 5
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
If `sqrt3 cos x − sin x = 1`, then general value of x is ______.
Select the correct option from the given alternatives:
In ΔABC, ac cos B - bc cos A = _______
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
Select the correct option from the given alternatives:
The value of cot (tan-12x + cot-12x) is
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Find the general solutions of the following equation:
`tan theta = - sqrt3`
Find the general solutions of the following equation:
`tan^2 theta = 3`
Find the general solutions of the following equation:
sin2 θ - cos2 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`
If `(sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C"))`, then show that a2, b2, c2 are in A.P.
State whether the following equation has a solution or not?
3 sin θ = 5
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
If tan-12x + tan-13x = `pi/4`, 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`.
Show that `cos^-1 sqrt3/2 + 2 sin^-1 sqrt3/2 = (5pi)/6`.
Show that `2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
The general solution of sec θ = `sqrt2` is
If tan-1 x + 2cot-1 x = `(5pi)/6`, then x is
Which of the following equations has no solution?
The value of `tan^-1 1/3 + tan^-1 1/5 + tan^-1 1/7 + tan^-1 1/8` is ______.
The value of sin 18° is ______.
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied, if ______.
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.
General solution of the equation sin 2x – sin 4x + sin 6x = 0 is ______.
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 of the equation tan2 x = 1 is ______.
The general solution of cot 4x = –1 is ______.
If `2sin^-1 3/7` = cos–1β, then find the value of β.
