Advertisements
Advertisements
Question
sin (tan–1 x), |x| < 1 is equal to ______.
Options
`x/(sqrt(1-x^2))`
`1/sqrt(1-x^2)`
`1/sqrt(1+x^2)`
`x/(sqrt(1+ x^2))`
Advertisements
Solution
sin (tan–1 x), |x| < 1 is equal to `bbunderline (x/(sqrt(1+ x^2)))`.
Explanation:
Let tan–1 x = θ
= x = tan θ, where `θ ∈ (- pi/2, pi/2)`
∴ sin (tan–1 x) = sin θ
Now, sin θ = `1/("cosec" θ)`
= `1/sqrt(1+cot^2θ)`
= `1/sqrt(1+ 1/tan^2θ)`
= `x/(sqrt(x^2 + 1)`
APPEARS IN
RELATED QUESTIONS
If `sin (sin^(−1)(1/5)+cos^(−1) x)=1`, then find the value of x.
Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,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.
Write the following function in the simplest form:
`tan^(-1) (sqrt(1+x^2) -1)/x`, x ≠ 0
Find the value of the following:
`tan^-1 [2 cos (2 sin^-1 1/2)]`
Find the value of `cot(tan^(-1) a + cot^(-1) a)`
if `sin(sin^(-1) 1/5 + cos^(-1) x) = 1` then find the value of x
if `tan^(-1) (x-1)/(x - 2) + tan^(-1) (x + 1)/(x + 2) = pi/4` then find the value of x.
`sin[pi/3 - sin^(-1) (-1/2)]` is equal to ______.
Prove `tan^(-1) 1/5 + tan^(-1) (1/7) + tan^(-1) 1/3 + tan^(-1) 1/8 = pi/4`
Prove that:
`tan^(-1) sqrtx = 1/2 cos^(-1) (1-x)/(1+x)`, x ∈ [0, 1]
Solve the following equation:
2 tan−1 (cos x) = tan−1 (2 cosec x)
Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`
Find the value, if it exists. If not, give the reason for non-existence
`tan^-1(sin(- (5pi)/2))`
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:
sin–1(2 cos2x – 1) + cos–1(1 – 2 sin2x) =
Evaluate `tan^-1(sin((-pi)/2))`.
Evaluate: `sin^-1 [cos(sin^-1 sqrt(3)/2)]`
Evaluate `cos[sin^-1 1/4 + sec^-1 4/3]`
Prove that `sin^-1 8/17 + sin^-1 3/5 = sin^-1 7/85`
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?
The value of the expression `tan (1/2 cos^-1 2/sqrt(5))` is ______.
If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.
The minimum value of sinx - cosx is ____________.
The value of `"tan"^-1 (1/2) + "tan"^-1 (1/3) + "tan"^-1 (7/8)` is ____________.
The value of expression 2 `"sec"^-1 2 + "sin"^-1 (1/2)`
If `"tan"^-1 (("x" - 1)/("x" + 2)) + "tan"^-1 (("x" + 1)/("x" + 2)) = pi/4,` then x is equal to ____________.
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:
The value of `"cos"^-1 ("cos" ((33pi)/5))` is ____________.
`"cos"^-1 (1/2)`
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 ____________.
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:
Domain and Range of tan-1 x = ________.
The value of `tan^-1 (x/y) - tan^-1 (x - y)/(x + y)` 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`
If sin–1x + sin–1y + sin–1z = π, show that `x^2 - y^2 - z^2 + 2yzsqrt(1 - x^2) = 0`
Solve:
sin–1 (x) + sin–1 (1 – x) = cos–1 x
