Advertisements
Advertisements
प्रश्न
Prove that:
`tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x))) = pi/4 - 1/2 cos^-1 x`, for `- 1/sqrt2 ≤ x ≤ 1`
[Hint: Put x = cos 2θ]
Advertisements
उत्तर
Put x = cos θ
∴ θ = cos–1 x
L.H.S. = `tan^-1 ((sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x)))`
= `tan^-1 ((sqrt(1 + cos θ) - sqrt(1 - cos θ))/(sqrt(1 + cos θ) + sqrt(1 - cos θ)))`
= `tan^-1 [(sqrt(2 cos^2(θ/2)) - sqrt(2 sin^2 (θ/2)))/(sqrt(2 cos^2 (θ/2)) + sqrt(2 sin^2 (θ/2)))]`
= `tan^-1 [(sqrt(2) cos (θ/2) - sqrt(2) sin (θ/2))/(sqrt(2) cos (θ/2) + sqrt(2) sin (θ/2))]`
= `tan^-1 [((sqrt(2) cos (θ/2))/(sqrt(2) cos (θ/2)) - (sqrt(2) sin (θ/2))/(sqrt(2) cos (θ/2)))/((sqrt(2) cos (θ/2))/(sqrt(2) cos (θ/2)) + (sqrt(2) sin (θ/2))/(sqrt(2) cos (θ/2)))]`
= `tan^-1 [(1 - tan(θ/2))/(1 + tan (θ/2))]`
= `tan^-1 [(tan pi/4 - tan (θ/2))/(1 + tan pi/4. tan (θ/2))] ....[∵ tan pi/4 =1]`
= `tan^-1 [tan (pi/4 - θ/2)]`
= `pi/4 - θ/2`
= `pi/4 - 1/2 cos^-1`x .....[∵ θ = cos–1 x]
∴ L.H.S. = R.H.S.
संबंधित प्रश्न
If `sin^-1(1-x) -2sin^-1x = pi/2` then x is
- -1/2
- 1
- 0
- 1/2
Find the principal value of the following:
`tan^(-1) (-sqrt3)`
Find the principal value of the following:
`cos^(-1) (-1/2)`
If sin−1 x = y, then ______.
`tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to ______.
Find the domain of the following function:
`f(x)=sin^-1x^2`
Find the domain of the following function:
`f(x) = sin^-1x + sinx`
Evaluate the following:
`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`
Evaluate the following:
`cot^-1{2cos(sin^-1 sqrt3/2)}`
Evaluate the following:
`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of sin `(A/2)`.
In ΔABC, if a = 18, b = 24, c = 30 then find the values of cos `A/2`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of sinA.
In ΔABC prove that `sin "A"/(2). sin "B"/(2). sin "C"/(2) = ["A(ΔABC)"]^2/"abcs"`
Find the principal value of the following: cos- 1`(-1/2)`
Evaluate the following:
`cos^-1(1/2) + 2sin^-1(1/2)`
Evaluate the following:
`tan^-1 sqrt(3) - sec^-1 (-2)`
Prove the following:
`cos^-1(3/5) + cos^-1(4/5) = pi/(2)`
Prove the following:
`tan^-1(1/2) + tan^-1(1/3) = pi/(4)`
In ΔABC, prove the following:
`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`
Find the principal solutions of the following equation:
sin 2θ = `− 1/(sqrt2)`
`tan^-1(tan (7pi)/6)` = ______
Evaluate cot(tan−1(2x) + cot−1(2x))
Prove that cot−1(7) + 2 cot−1(3) = `pi/4`
Find the principal value of the following:
`sec^-1 (-sqrt2)`
Evaluate: sin`[1/2 cos^-1 (4/5)]`
Find the principal value of `cos^-1 sqrt(3)/2`
`sin^-1x + sin^-1 1/x + cos^-1x + cos^-1 1/x` = ______
The value of 2 `cot^-1 1/2 - cot^-1 4/3` is ______
`sin^2(sin^-1 1/2) + tan^2 (sec^-1 2) + cot^2(cosec^-1 4)` = ______.
The principal value of `sin^-1 (sin (3pi)/4)` is ______.
`tan[2tan^-1 (1/3) - pi/4]` = ______.
In a triangle ABC, ∠C = 90°, then the value of `tan^-1 ("a"/("b + c")) + tan^-1("b"/("c + a"))` is ______.
`cos(2sin^-1 3/4+cos^-1 3/4)=` ______.
`(sin^-1(-1/2) + tan^-1(-1/sqrt(3)))/(sec^-1 (-2/sqrt(3)) + cos^-1(1/sqrt(2))` = ______.
The value of `sin^-1[cos(pi/3)] + sin^-1[tan((5pi)/4)]` is ______.
The domain of y = cos–1(x2 – 4) is ______.
The domain of the function defined by f(x) = sin–1x + cosx is ______.
Prove that `cot(pi/4 - 2cot^-1 3)` = 7
If 2 tan–1(cos θ) = tan–1(2 cosec θ), then show that θ = π 4, where n is any integer.
Show that `cos(2tan^-1 1/7) = sin(4tan^-1 1/3)`
All trigonometric functions have inverse over their respective domains.
`("cos" 8° - "sin" 8°)/("cos" 8° + "sin" 8°)` is equal to ____________.
If sin-1 x – cos-1 x `= pi/6,` then x = ____________.
`"tan"^-1 (sqrt3)`
`2 "tan"^-1 ("cos x") = "tan"^-1 (2 "cosec x")`
`"sin" ["cot"^-1 {"cos" ("tan"^-1 "x")}] =` ____________.
The value of `"cos"^-1 ("cos" ((33 pi)/5))` is ____________.
If `"x" in (- pi/2, pi/2), "then the value of tan"^-1 ("tan x"/4) + "tan"^-1 ((3 "sin" 2 "x")/(5 + 3 "cos" 2 "x"))` is ____________.
If tan-1 x – tan-1 y = tan-1 A, then A is equal to ____________.
`"cos" ["tan"^-1 {"sin" ("cot"^-1 "x")}]` is equal to ____________.
`2"tan"^-1 ("cos x") = "tan"^-1 (2 "cosec x")`
`"cos"^-1 ("cos" ((7pi)/6))` is equal to ____________.
If a = `(2sin theta)/(1 + costheta + sintheta)`, then `(1 + sintheta - costheta)/(1 + sintheta)` is
If A = `[(cosx, sinx),(-sinx, cosx)]`, then A1 A–1 is
If `(-1)/sqrt(2) ≤ x ≤ 1/sqrt(2)` then `sin^-1 (2xsqrt(1 - x^2))` is equal to
Domain and Rariges of cos–1 is:-
What is the principal value of cosec–1(2).
`2tan^-1 (cos x) = tan^-1 (2"cosec" x)`, then 'x' will be equal to
what is the value of `cos^-1 (cos (13pi)/6)`
Value of `sin(pi/3 - sin^1 (- 1/2))` is equal to
If f(x) = x5 + 2x – 3, then (f–1)1 (–3) = ______.
If θ = `sin^-1((2x)/(1 + x^2)) + cos^-1((1 - x^2)/(1 + x^2))`, for `x ≥ 3/2` then the absolute value of `((cosθ + tanθ + 4)/secθ)` is ______.
Consider f(x) = sin–1[2x] + cos–1([x] – 1) (where [.] denotes greatest integer function.) If domain of f(x) is [a, b) and the range of f(x) is {c, d} then `a + b + (2d)/c` is equal to ______. (where c < d)
cos–1(cos10) is equal to ______.
If x ∈ R – {0}, then `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2)))`
If y = `tan^-1 (sqrt(1 + x^2) - sqrt(1 - x^2))/(sqrt(1 + x^2) + sqrt(1 - x^2))`, then `dy/dx` is equal to ______.
If cos–1 x > sin–1 x, then ______.
`sin[π/3 + sin^-1 (1/2)]` is equal to ______.
The value of `tan(cos^-1 4/5 + tan^-1 2/3)` is ______.
Solve for x:
5tan–1x + 3cot–1x = 2π
If tan 4θ = `tan(2/θ)`, then the general value of θ is ______.
