Advertisements
Advertisements
Question
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Advertisements
Solution
Let x = sin θ.
Then, sin−1 x = θ.
We have,
R.H.S = sin−1 (3x – 4x3) = sin−1 (3 sin θ – 4 sin3θ)
= sin−1 (sin 3θ) = sin−1 (3 sin θ – 4 sin3θ)
= 3θ = sin−1 (3 sin θ – 4 sin3θ)
= 3 sin−1 x = sin−1 (3 sin θ – 4 sin3θ)
R.H.S = L.H.S
APPEARS IN
RELATED QUESTIONS
If `sin (sin^(−1) 1/5+cos^(−1) x)=1`, then find the value of x.
If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.
If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.
Write the function in the simplest form: `tan^(-1) 1/(sqrt(x^2 - 1)), |x| > 1`
Write the following function in the simplest form:
`tan^(-1) (sqrt((1 - cos x)/(1 + cos x)))`, 0 < x < π
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`
Find the value of the given expression.
`tan^(-1) (tan (3pi)/4)`
Prove that `cos^(-1) 4/5 + cos^(-1) 12/13 = cos^(-1) 33/65`.
Prove that `cot^(-1) ((sqrt(1 + sin x) + sqrt(1 - sinx))/(sqrt(1 + sin x) - sqrt(1 - sinx))) = x/2, x ∈ (0, pi/4)`.
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)`.
Find the value, if it exists. If not, give the reason for non-existence
`tan^-1(sin(- (5pi)/2))`
Prove that `sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
Solve: `sin^-1 5/x + sin^-1 12/x = pi/2`
Solve: `2tan^-1 (cos x) = tan^-1 (2"cosec" x)`
Choose the correct alternative:
If |x| ≤ 1, then `2tan^-1x - sin^-1 (2x)/(1 + x^2)` is equal to
Choose the correct alternative:
The equation tan–1x – cot–1x = `tan^-1 (1/sqrt(3))` has
Evaluate: `tan^-1 sqrt(3) - sec^-1(-2)`.
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"))]`
If cos–1α + cos–1β + cos–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.
The value of cot–1(–x) for all x ∈ R in terms of cot–1x is ______.
The value of `"tan"^ -1 (3/4) + "tan"^-1 (1/7)` is ____________.
The value of sin (2tan-1 (0.75)) is equal to ____________.
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 =` ____________.
If sin `("sin"^-1 1/5 + "cos"^-1 "x") = 1,` then the value of x is ____________.
If `6"sin"^-1 ("x"^2 - 6"x" + 8.5) = pi,` then x is equal to ____________.
`"tan"^-1 (sqrt3)`
`tan^-1 1/2 + tan^-1 2/11` is equal to
The Simplest form of `cot^-1 (1/sqrt(x^2 - 1))`, |x| > 1 is
What is the value of cos (sec–1x + cosec–1x), |x| ≥ 1
`50tan(3tan^-1(1/2) + 2cos^-1(1/sqrt(5))) + 4sqrt(2) tan(1/2tan^-1(2sqrt(2)))` is equal to ______.
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
The value of cosec `[sin^-1((-1)/2)] - sec[cos^-1((-1)/2)]` is equal to ______.
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 ______.
`sin (tan^-1 4/5 + tan^-1 4/3 + tan^-1 1/9 - tan^-1 1/7)` is equal to ______.
