Advertisements
Advertisements
प्रश्न
Prove that `cos^(-1) 12/13 + sin^(-1) 3/5 = sin^(-1) 56/65`.
Advertisements
उत्तर
Let `sin^-1 3/5 = x`.
Then, `sin x = 3/5`
⇒ `cos x = sqrt(1 - (3/5)^2`
= `sqrt(16/25)`
= `4/5`
∴ `tan x = 3/4` ⇒ `x = tan^-1 3/4`
∴ `sin^-1 3/5 = tan^-1 3/4` ...(1)
Now, let `cos^-1 12/13 = y`.
Then, `cos y = 12/13` ⇒ `sin y = 5/13`.
∴ `tan y = 5/12` ⇒ `y = tan^-1 5/12`
∴ `cos^-1 12/13 = tan^-1 5/12` ...(2)
Let `sin^-1 56/65 = z`.
Then, `sin z = 56/65` ⇒ `cos z = 33/65`.
∴ `tan z = 56/33` ⇒ `z = tan^-1 56/33`
∴ `sin^-1 56/65 = tan^-1 56/33` ...(3)
Now, we have:
L.H.S. = `cos^-1 12/13 + sin^-1 3/5`
= `tan^-1 5/12 + tan^-1 3/4` ...[Using (1) and (2)]
= `tan^-1 (5/12 + 3/4)/(1 - 5/12 * 3/4)` ...`[tan^-1x + tan^-1y = tan^-1 (x + y)/(1 - xy)]`
= `tan^-1 (20 + 36)/(48 - 15)`
= `tan^-1 56/33`
= `sin^-1 56/65` = R.H.S. ...[Using (3)]
APPEARS IN
संबंधित प्रश्न
Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `
Write the following function in the simplest form:
`tan^(-1) (sqrt(1 + x^2) - 1)/x, x ≠ 0`
Write the following function in the simplest form:
`tan^(-1) (sqrt((1 - cos x)/(1 + cos x)))`, 0 < x < π
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
Prove that `sin^(-1) 8/17 + sin^(-1) 3/5 = tan^(-1) 77/36`.
Prove `(9pi)/8 - 9/4 sin^(-1) 1/3 = 9/4 sin^(-1) (2sqrt2)/3`
sin–1 (1 – x) – 2 sin–1 x = `pi/2`, then x is equal to ______.
Solve `tan^(-1) - tan^(-1) (x - y)/(x+y)` is equal to
(A) `pi/2`
(B). `pi/3`
(C) `pi/4`
(D) `(-3pi)/4`
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Solve the following equation for x: `cos (tan^(-1) x) = sin (cot^(-1) 3/4)`
If tan-1 x - cot-1 x = tan-1 `(1/sqrt(3)),`x> 0 then find the value of x and hence find the value of sec-1 `(2/x)`.
Find the value, if it exists. If not, give the reason for non-existence
`sin^-1 [sin 5]`
Find the value of the expression in terms of x, with the help of a reference triangle
sin (cos–1(1 – x))
Find the value of the expression in terms of x, with the help of a reference triangle
cos (tan–1 (3x – 1))
Find the value of `sin^-1[cos(sin^-1 (sqrt(3)/2))]`
Prove that `tan^-1x + tan^-1y + tan^-1z = tan^-1[(x + y + z - xyz)/(1 - xy - yz - zx)]`
Solve: `tan^-1x = cos^-1 (1 - "a"^2)/(1 + "a"^2) - cos^-1 (1 - "b"^2)/(1 + "b"^2), "a" > 0, "b" > 0`
If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.
Show that `tan(1/2 sin^-1 3/4) = (4 - sqrt(7))/3` and justify why the other value `(4 + sqrt(7))/3` is ignored?
If 3 tan–1x + cot–1x = π, then x equals ______.
If `sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2))`. where a, x ∈ ] 0, 1, then the value of x is ______.
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.
The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.
The maximum value of sinx + cosx is ____________.
Solve for x : `"sin"^-1 2 "x" + sin^-1 3"x" = pi/3`
`"cot" (pi/4 - 2 "cot"^-1 3) =` ____________.
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
`"tan"^-1 1/3 + "tan"^-1 1/5 + "tan"^-1 1/7 = "tan"^-1 1/8 =` ____________.
The value of `"tan"^-1 (3/4) + "tan"^-1 (1/7)` is ____________.
`"sin"^-1 (1 - "x") - 2 "sin"^-1 "x" = pi/2`
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
`"sin"^-1 (1/sqrt2)`
The value of `tan^-1 (x/y) - tan^-1 (x - y)/(x + y)` is equal to
Find the value of `sin^-1 [sin((13π)/7)]`
If `tan^-1 ((x - 1)/(x + 1)) + tan^-1 ((2x - 1)/(2x + 1)) = tan^-1 (23/36)` = then prove that 24x2 – 23x – 12 = 0
Solve:
sin–1 (x) + sin–1 (1 – x) = cos–1 x
`cos^(−1)(1/2) + sin^(−1)(1) + tan^(−1) 1/sqrt3` is equal to ______.
