Advertisements
Advertisements
प्रश्न
Prove that `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/((1 + x^2) - sqrt(1 - x^2))) = pi/2 + 1/2 cos^-1x^2`
उत्तर
L.H.S. `tan^-1 [(sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2 - sqrt(1 - x^2)))]`
Put x2 = cos θ
∴ θ = `cos^-1 x^2`
⇒ `tan^-1 [(sqrt(1 + cos theta) + sqrt(1 - cos theta))/(sqrt(1 + cos theta) - sqrt(1 - cos theta))]`
⇒ `tan^-1 [sqrt(2cos^2 theta/2 + sqrt(2sin^2 theta/2))/(sqrt(2cos^2 theta/2 - sqrt(2sin^2 theta/2)))]` ......`[(because 1 + cos theta = 2 cos^2 theta/2),(1 - cos theta = 2 sin^2 theta/2)]`
⇒ `tan^-1 [(cos theta/2 + sin theta/2),(cos theta/2 - sin theta/2)]`
⇒ `tan^-1 [(1 + tan theta/2),(1 - tan theta/2)]` ......[Dividing the Nr. and Den. by cos θ/2]
⇒ `tan [tan(pi/4 theta/2)]` ......`[because (1 + tan theta)/(1 - tan theta) = tan(pi/4 + theta)]`
⇒ `pi/4 + theta/2`
⇒ `pi/4 + 1/2 cos^-1 x^2` R.H.S. ......[Putting θ = cos–1x2]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
Prove the following:
`3sin^(-1) x = sin^(-1)(3x - 4x^3), x in [-1/2, 1/2]`
Prove the following:
`3cos^(-1) x = cos^(-1)(4x^3 - 3x), x in [1/2, 1]`
Prove `tan^(-1) 1/5 + tan^(-1) (1/7) + tan^(-1) 1/3 + tan^(-1) 1/8 = pi/4`
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
Solve: tan-1 4 x + tan-1 6x `= π/(4)`.
Find the value, if it exists. If not, give the reason for non-existence
`sin^-1 (cos pi)`
Prove that `tan^-1x + tan^-1 (2x)/(1 - x^2) = tan^-1 (3x - x^3)/(1 - 3x^2), |x| < 1/sqrt(3)`
Solve: `cot^-1 x - cot^-1 (x + 2) = pi/12, x > 0`
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:
The equation tan–1x – cot–1x = `tan^-1 (1/sqrt(3))` has
Evaluate tan (tan–1(– 4)).
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 `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 ______.
The maximum value of sinx + cosx is ____________.
`"cot" ("cosec"^-1 5/3 + "tan"^-1 2/3) =` ____________.
The value of `"tan"^-1 (3/4) + "tan"^-1 (1/7)` is ____________.
If `"tan"^-1 2 "x + tan"^-1 3 "x" = pi/4`, then x is ____________.
`"tan"^-1 1/3 + "tan"^-1 1/5 + "tan"^-1 1/7 + "tan"^-1 1/8 =` ____________.
The value of `"cos"^-1 ("cos" ((33pi)/5))` is ____________.
`"tan"^-1 1 + "cos"^-1 ((-1)/2) + "sin"^-1 ((-1)/2)`
`"sin"^-1 ((-1)/2)`
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 ∠CAB = ________.
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 ______.
The Simplest form of `cot^-1 (1/sqrt(x^2 - 1))`, |x| > 1 is
What is the simplest form of `tan^-1 sqrt(1 - x^2 - 1)/x, x ≠ 0`
The set of all values of k for which (tan–1 x)3 + (cot–1 x)3 = kπ3, x ∈ R, is the internal ______.
