Advertisements
Advertisements
Question
Prove the following:
`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)`
Solution
L.H.S. = `tan^-1["cosθ + sinθ"/"cosθ - sinθ"]`
= `tan^-1 [((cosθ)/(cosθ) + (sinθ)/(cosθ))/((cosθ)/(cosθ) - (sinθ)/(cosθ))]`
= `tan^-1((1 + tanθ)/(1 - tanθ))`
= `tan^-1[(tan pi/4 + tanθ)/(1 - tan pi/4 tan θ)]`
= `tan^-1[tan(pi/4 + θ)]`
= `pi/(4) + θ` ...[∵ tan–1(tan θ) = θ]
= R.H.S.
APPEARS IN
RELATED QUESTIONS
If `sin^-1(1-x) -2sin^-1x = pi/2` then x is
- -1/2
- 1
- 0
- 1/2
Show that `2sin^-1(3/5) = tan^-1(24/7)`
Show that:
`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`
Find the principal value of `sec^(-1) (2/sqrt(3))`
Find the principal value of `cot^(-1) (sqrt3)`
Find the principal value of `cos^(-1) (-1/sqrt2)`
Find the value of the following:
`cos^(-1) (1/2) + 2 sin^(-1)(1/2)`
Find the principal value of `sin^-1(1/sqrt2)`
Find the domain of the following function:
`f(x)sin^-1sqrt(x^2-1)`
If `sin^-1 x + sin^-1 y+sin^-1 z+sin^-1 t=2pi` , then find the value of x2 + y2 + z2 + t2
Evaluate the following:
`tan^-1(-1/sqrt3)+tan^-1(-sqrt3)+tan^-1(sin(-pi/2))`
Find the set of values of `cosec^-1(sqrt3/2)`
Find the domain of `f(x)=cotx+cot^-1x`
Evaluate the following:
`cot^-1{2cos(sin^-1 sqrt3/2)}`
Evaluate the following:
`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of tan `A/2`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of A(ΔABC)
In ΔABC, if a = 18, b = 24, c = 30 then find the values of sinA
Find the principal value of the following: cosec- 1(2)
Evaluate the following:
`tan^-1(1) + cos^-1(1/2) + sin^-1(1/2)`
Evaluate the following:
`cos^-1(1/2) + 2sin^-1(1/2)`
Evaluate the following:
`tan^-1 sqrt(3) - sec^-1 (-2)`
Prove the following:
`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`
Prove the following:
`sin^-1(-1/2) + cos^-1(-sqrt(3)/2) = cos^-1(-1/2)`
Prove the following:
`2tan^-1(1/3) = tan^-1(3/4)`
In ΔABC, prove the following:
`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`
sin−1x − cos−1x = `pi/6`, then x = ______
If `sin(sin^-1(1/5) + cos^-1(x))` = 1, then x = ______
Evaluate `cos[pi/6 + cos^-1 (- sqrt(3)/2)]`
If tan−1x + tan−1y + tan−1z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1
Find the principal value of the following:
`sec^-1 (-sqrt2)`
Prove that:
2 tan-1 (x) = `sin^-1 ((2x)/(1 + x^2))`
Solve `tan^-1 2x + tan^-1 3x = pi/4`
Solve: tan-1 (x + 1) + tan-1 (x – 1) = `tan^-1 (4/7)`
Evaluate: sin`[1/2 cos^-1 (4/5)]`
Express `tan^-1 ((cos x - sin x)/(cos x + sin x))`, 0 < x < π in the simplest form.
Find the principal value of `cos^-1 sqrt(3)/2`
`sin^-1x + sin^-1 1/x + cos^-1x + cos^-1 1/x` = ______
lf `sqrt3costheta + sintheta = sqrt2`, then the general value of θ is ______
If `sin^-1(x/13) + cosec^-1(13/12) = pi/2`, then the value of x is ______
In Δ ABC, with the usual notations, if sin B sin C = `"bc"/"a"^2`, then the triangle is ______.
The value of 2 `cot^-1 1/2 - cot^-1 4/3` is ______
If `sin^-1 3/5 + cos^-1 12/13 = sin^-1 P`, then P is equal to ______
The principal value of `sin^-1 (sin (3pi)/4)` is ______.
If sin `(sin^-1 1/3 + cos^-1 x) = 1`, then the value of x is ______.
`tan[2tan^-1 (1/3) - pi/4]` = ______.
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then θ = ______
The value of `sin^-1(cos (53pi)/5)` is ______
The value of `cos(pi/4 + x) - cos(pi/4 - x)` is ______.
The value of `sin^-1[cos(pi/3)] + sin^-1[tan((5pi)/4)]` is ______.
The domain of the function y = sin–1 (– x2) is ______.
Show that `sin^-1 5/13 + cos^-1 3/5 = tan^-1 63/16`
Prove that `tan^-1 1/4 + tan^-1 2/9 = sin^-1 1/sqrt(5)`
All trigonometric functions have inverse over their respective domains.
If `"cos"^-1 "x + sin"^-1 "x" = pi`, then the value of x is ____________.
If tan-1 3 + tan-1 x = tan-1 8, then x = ____________.
`"sin"^-1 (1/sqrt2)`
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
The equation 2cos-1 x + sin-1 x `= (11pi)/6` has ____________.
`sin[π/3 - sin^-1 (-1/2)]` is equal to:
If `"cot"^-1 (sqrt"cos" alpha) - "tan"^-1 (sqrt "cos" alpha) = "x",` then sinx is equal to ____________.
`"cos"^-1 ("cos" ((7pi)/6))` is equal to ____________.
The equation of the tangent to the curve given by x = a sin3t, y = bcos3t at a point where t = `pi/2` is
What is the principal value of cosec–1(2).
`2tan^-1 (cos x) = tan^-1 (2"cosec" x)`, then 'x' will be equal to
What is the values of `cos^-1 (cos (7pi)/6)`
`lim_(n→∞)tan{sum_(r = 1)^n tan^-1(1/(1 + r + r^2))}` is equal to ______.
If tan–1 2x + tan–1 3x = `π/4`, then x = ______.
Derivative of `tan^-1(x/sqrt(1 - x^2))` with respect sin–1(3x – 4x3) is ______.
`(tan^-1 (sqrt(3)) - sec^-1(-2))/("cosec"^-1(-sqrt(2)) + cos^-1(-1/2))` is equal to ______.
If cos–1 x > sin–1 x, then ______.
