Advertisements
Advertisements
Question
Prove that `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sinx))/(sqrt(1 + sin x) - sqrt(1 - sinx))) = x/2, x ∈ (0, pi/4)`.
Advertisements
Solution
Consider `(sqrt(1 + sinx) + sqrt(1 - sin x))/(sqrt(1 + sinx) - sqrt(1 - sinx))`
= `((sqrt(1 + sinx) + sqrt(1 - sinx))^2)/((sqrt(1 + sin x))^2 - (sqrt(1 - sin x))^2)` ...(By rationalizing)
= `((1 + sinx) + (1 - sinx) + 2sqrt((1 + sinx)(1 - sinx)))/(1 + sinx - 1 + sinx)`
= `(2(1 + sqrt(1 - sin^2x)))/(2sinx)`
= `(1 + cosx)/(sin x)`
= `(2 cos^2 x/2)/(2sin x/2 cos x/2)`
= `cot x/2`
∴ L.H.S = `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sinx))/(sqrt(1 + sin x) - sqrt(1 - sinx)))`
= `cot^(-1) (cot x/2)`
= `x/2` = R.H.S.
APPEARS IN
RELATED QUESTIONS
Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `
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`
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Write the following function in the simplest form:
`tan^(-1) ((3a^2 x - x^3)/(a^3 - 3ax^2)), a > 0; (-a)/sqrt(3) < x < a/sqrt(3)`
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
Find the value of the given expression.
`sin^(-1) (sin (2pi)/3)`
Find the value of the given expression.
`tan^(-1) (tan (3pi)/4)`
Prove `tan^(-1) 1/5 + tan^(-1) (1/7) + tan^(-1) 1/3 + tan^(-1) 1/8 = pi/4`
sin–1 (1 – x) – 2 sin–1 x = `pi/2`, then x is equal to ______.
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 cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.
Find the value, if it exists. If not, give the reason for non-existence
`sin^-1 [sin 5]`
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:
If |x| ≤ 1, then `2tan^-1x - sin^-1 (2x)/(1 + x^2)` is equal to
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 `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 ______.
`"cot" (pi/4 - 2 "cot"^-1 3) =` ____________.
`"tan"^-1 1 + "cos"^-1 ((-1)/2) + "sin"^-1 ((-1)/2)`
`"sin" {2 "cos"^-1 ((-3)/5)}` is equal to ____________.
The value of cot-1 9 + cosec-1 `(sqrt41/4)` is given by ____________.
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:
Solve for x : `{"x cos" ("cot"^-1 "x") + "sin" ("cot"^-1 "x")}^2` = `51/50
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 ∠DAB = ________.
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 = ________.
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 simplest form of `tan^-1 sqrt(1 - x^2 - 1)/x, x ≠ 0`
Find the value of `sin^-1 [sin((13π)/7)]`
Find the value of `tan^-1 [2 cos (2 sin^-1 1/2)] + tan^-1 1`.
If \[\tan^{-1}\left(\frac{x}{2}\right)+\tan^{-1}\left(\frac{y}{2}\right)+\tan^{-1}\left(\frac{z}{2}\right)=\frac{\pi}{2}\] then xy + yz + zx =
Principal value of `"cosec"^(−1)((−2)/sqrt3)` is equal to ______.
