Advertisements
Advertisements
प्रश्न
If cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.
Advertisements
उत्तर
cos-1 x + cos -1 y + cos -1 z = π
cos-1 x + cos -1 y = π - cos -1 z
cos-1 `(xy - sqrt(1 - x^2) sqrt(1 -y^2 ))` = π - cos -1 z
`xy - sqrt(1 - x^2) sqrt(1 -y^2)` = cos ( π - cos -1 z)
`xy - sqrt(1 - x^2) sqrt(1 -y^2)` = - cos(cos-1 z)
xy - `sqrt(1 - x^2) sqrt(1 -y^2) = -z`
`xy + z = sqrt(1 - x^2) sqrt(1 - y^2)`
Squaring both sides, we have
(xy + z)2 = (1 - x2) (1- y2)
x2y2 + z2 + 2xyz = 1 - x2 - y2 + x2y2
x2 + y2 + z2 + 2xyz = 1
APPEARS IN
संबंधित प्रश्न
Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`
Prove that:
`tan^(-1)""1/5+tan^(-1)""1/7+tan^(-1)""1/3+tan^(-1)""1/8=pi/4`
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt3 < x < a/sqrt3`
Find the value of the following:
`tan 1/2 [sin^(-1) (2x)/(1+ x^2) + cos^(-1) (1-y^2)/(1+y^2)]`, |x| < 1, y > 0 and xy < 1
Prove `(9pi)/8 - 9/4 sin^(-1) 1/3 = 9/4 sin^(-1) (2sqrt2)/3`
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Solve for x : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
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)`.
Prove that `sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
Choose the correct alternative:
If |x| ≤ 1, then `2tan^-1x - sin^-1 (2x)/(1 + x^2)` is equal to
Choose the correct alternative:
sin(tan–1x), |x| < 1 is equal to
Evaluate: `tan^-1 sqrt(3) - sec^-1(-2)`.
Prove that `2sin^-1 3/5 - tan^-1 17/31 = pi/4`
Prove that `sin^-1 8/17 + sin^-1 3/5 = sin^-1 7/85`
The maximum value of sinx + cosx is ____________.
The value of cot `("cosec"^-1 5/3 + "tan"^-1 2/3)` is ____________.
The value of expression 2 `"sec"^-1 2 + "sin"^-1 (1/2)`
`"tan"^-1 1/3 + "tan"^-1 1/5 + "tan"^-1 1/7 = "tan"^-1 1/8 =` ____________.
sin (tan−1 x), where |x| < 1, is equal to:
If x = a sec θ, y = b tan θ, then `("d"^2"y")/("dx"^2)` at θ = `π/6` is:
If `"tan"^-1 2 "x + tan"^-1 3 "x" = pi/4`, then x is ____________.
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:
𝐴' Is another viewer standing on the same line of observation across the road. If the width of the road is 5 meters, then the difference between ∠CAB and ∠CA'B is ______.
What is the value of cos (sec–1x + cosec–1x), |x| ≥ 1
Find the value of `sin^-1 [sin((13π)/7)]`
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
Write the following function in the simplest form:
`tan^-1 ((cos x - sin x)/(cos x + sin x)), (-pi)/4 < x < (3 pi)/4`
