Advertisements
Advertisements
प्रश्न
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
Advertisements
उत्तर
Here, we have y = `(x sin^-1 x)/sqrt(1 -x^2)`
y `sqrt(1 -x^2)` = x sin-1 x ....(i)
Differentiate both sides w.r.t. x, we have
`y ((-2x))/(2sqrt(1 -x^2)) + sqrt(1 - x^2) (dy)/(dx) = x (1)/sqrt(1 -x^2) + sin^-1 x`
`- xy + (1 - x^2) (dy)/(dx) = x + sqrt(1 - x^2) sin^-1 x`
`- xy + (1 - x^2) (dy)/(dx) = x + sqrt(1 - x^2) . (y)/(x) sqrt(1 -x^2) ...[ ∵ sin^-1 x = (y)/(x) sqrt(1 - x^2) , "using" (i) ]`
`- xy + (1 - x^2) (dy)/(dx) = x + (y)/(x) (1 - x^2)`
`- xy + (1 - x^2) (dy)/(dx) = x + (y)/(x) - yx`
`(1 - x^2) (dy)/(dx) = x + (y)/(x) `
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.
Prove that `sin^(-1) 8/17 + sin^(-1) 3/5 = tan^(-1) 77/36`.
Solve the following equation:
2 tan−1 (cos x) = tan−1 (2 cosec x)
sin (tan–1 x), |x| < 1 is equal to ______.
Find the value of `tan(sin^-1 3/5 + cot^-1 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`
Choose the correct alternative:
If `sin^-1x + sin^-1y = (2pi)/3` ; then `cos^-1x + cos^-1y` is equal to
Choose the correct alternative:
`sin^-1 (tan pi/4) - sin^-1 (sqrt(3/x)) = pi/6`. Then x is a root of the equation
Choose the correct alternative:
If `sin^-1x + cot^-1 (1/2) = pi/2`, then x is equal to
Prove that `2sin^-1 3/5 - tan^-1 17/31 = pi/4`
If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.
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"))]`
The value of the expression `tan (1/2 cos^-1 2/sqrt(5))` is ______.
The minimum value of sinx - cosx is ____________.
The value of expression 2 `"sec"^-1 2 + "sin"^-1 (1/2)`
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
If x = a sec θ, y = b tan θ, then `("d"^2"y")/("dx"^2)` at θ = `π/6` is:
Simplest form of `tan^-1 ((sqrt(1 + cos "x") + sqrt(1 - cos "x"))/(sqrt(1 + cos "x") - sqrt(1 - cos "x")))`, `π < "x" < (3π)/2` is:
`"cos"^-1["cos"(2"cot"^-1(sqrt2 - 1))]` = ____________.
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
The value of `"cos"^-1 ("cos" ((33pi)/5))` is ____________.
If `6"sin"^-1 ("x"^2 - 6"x" + 8.5) = pi,` 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 ____________.
The Government of India is planning to fix a hoarding board at the face of a building on the road of a busy market for awareness on COVID-19 protocol. Ram, Robert and Rahim are the three engineers who are working on this project. “A” is considered to be a person viewing the hoarding board 20 metres away from the building, standing at the edge of a pathway nearby. Ram, Robert and Rahim suggested to the firm to place the hoarding board at three different locations namely C, D and E. “C” is at the height of 10 metres from the ground level. For viewer A, the angle of elevation of “D” is double the angle of elevation of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer. Look at the figure given and based on the above information answer the following:
Measure of ∠EAB = ________.
Find the value of `cos^-1 (1/2) + 2sin^-1 (1/2) ->`:-
The set of all values of k for which (tan–1 x)3 + (cot–1 x)3 = kπ3, x ∈ R, is the internal ______.
