Advertisements
Advertisements
प्रश्न
Prove that `tan^(-1) 63/16 = sin^(-1) 5/13 + cos^(-1) 3/5`.
Advertisements
उत्तर
Let `sin^(-1) 5/13 = x`.
Then, `sin x = 5/13` ⇒ `cos x = 12/13`.
∴ `tan x = 5/12` ⇒ `x = tan^-1 5/12`
∴ `sin^-1 5/13 = tan^-1 5/12` ...(1)
Let `cos^-1 3/5 = y`.
Then, `cos y = 3/5` ⇒ `sin y = 4/5`.
∴ `tan y = 4/3` ⇒ `y = tan^-1 4/3`
∴ `cos^-1 3/5 = tan^-1 4/3` ...(2)
Using (1) and (2), we have
R.H.S. = `sin^-1 5/13 + cos^-1 3/5`
= `tan^-1 5/12 + tan^-1 4/3`
= `tan^-1 ((5/12 + 4/3)/(1 - 5/12 xx 4/3))` ...`[tan^-1x + tan^-1y = tan^-1 (x + y)/(1 - xy)]`
= `tan^-1 ((15 + 48)/(36 - 20))`
= `tan^-1 63/16`
= L.H.S.
APPEARS IN
संबंधित प्रश्न
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Prove the following:
3cos–1x = cos–1 (4x3 – 3x), `x ∈ [1/2, 1]`
Prove `2 tan^(-1) 1/2 + tan^(-1) 1/7 = tan^(-1) 31/17`
Write the function in the simplest form: `tan^(-1) 1/(sqrt(x^2 - 1)), |x| > 1`
Write the following function in the simplest form:
`tan^(-1) x/(sqrt(a^2 - x^2)), |x| < a`
if `sin(sin^(-1) 1/5 + cos^(-1) x) = 1` then find the value of x
Find the value of the given expression.
`sin^(-1) (sin (2pi)/3)`
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`
Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1 ("x")/(2), "x">0.`
Prove that `sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
Prove that `tan^-1x + tan^-1y + tan^-1z = tan^-1[(x + y + z - xyz)/(1 - xy - yz - zx)]`
Solve: `2tan^-1 (cos x) = tan^-1 (2"cosec" x)`
Find the number of solutions of the equation `tan^-1 (x - 1) + tan^-1x + tan^-1(x + 1) = tan^-1(3x)`
Choose the correct alternative:
`tan^-1 (1/4) + tan^-1 (2/9)` is equal to
Choose the correct alternative:
sin(tan–1x), |x| < 1 is equal to
Prove that cot–17 + cot–18 + cot–118 = cot–13
If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.
Evaluate `cos[cos^-1 ((-sqrt(3))/2) + pi/6]`
Prove that `sin^-1 8/17 + sin^-1 3/5 = sin^-1 7/85`
If a1, a2, a3,...,an is an arithmetic progression with common difference d, then evaluate the following expression.
`tan[tan^-1("d"/(1 + "a"_1 "a"_2)) + tan^-1("d"/(21 + "a"_2 "a"_3)) + tan^-1("d"/(1 + "a"_3 "a"_4)) + ... + tan^-1("d"/(1 + "a"_("n" - 1) "a""n"))]`
If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.
The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.
`"sin" {2 "cos"^-1 ((-3)/5)}` is equal to ____________.
The value of expression 2 `"sec"^-1 2 + "sin"^-1 (1/2)`
`"cot" ("cosec"^-1 5/3 + "tan"^-1 2/3) =` ____________.
If sin `("sin"^-1 1/5 + "cos"^-1 "x") = 1,` then the value of x is ____________.
`"cos"^-1["cos"(2"cot"^-1(sqrt2 - 1))]` = ____________.
If `6"sin"^-1 ("x"^2 - 6"x" + 8.5) = pi,` then x is equal to ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
If `"sin"^-1 (1 - "x") - 2 "sin"^-1 ("x") = pi/2,` then x is equal to ____________.
If `3 "sin"^-1 ((2"x")/(1 + "x"^2)) - 4 "cos"^-1 ((1 - "x"^2)/(1 + "x"^2)) + 2 "tan"^-1 ((2"x")/(1 - "x"^2)) = pi/3` then x is equal to ____________.
`50tan(3tan^-1(1/2) + 2cos^-1(1/sqrt(5))) + 4sqrt(2) tan(1/2tan^-1(2sqrt(2)))` is equal to ______.
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
Solve for x: `sin^-1(x/2) + cos^-1x = π/6`
Solve:
sin–1 (x) + sin–1 (1 – x) = cos–1 x
