Advertisements
Advertisements
प्रश्न
Prove the following:
`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)`
उत्तर
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
संबंधित प्रश्न
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)`
Find the principal value of `tan^(-1) (-sqrt3)`
Find the principal value of `cos^(-1) (-1/2)`
Find the principal value of tan−1 (−1)
Find the principal value of `cosec^(-1)(-sqrt2)`
`tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to ______.
Find the value of the following:
`tan^(-1) (tan (7x)/6)`
Find the principal value of `sin^-1(1/sqrt2)`
`sin^-1 1/2-2sin^-1 1/sqrt2`
Find the domain of the following function:
`f(x)=sin^-1x^2`
Evaluate the following:
`tan^-1(tan (5pi)/6)+cos^-1{cos((13pi)/6)}`
Find the set of values of `cosec^-1(sqrt3/2)`
Evaluate the following:
`cot^-1 1/sqrt3-\text(cosec)^-1(-2)+sec^-1(2/sqrt3)`
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 cosA
In ΔABC prove that `(b + c - a) tan "A"/(2) = (c + a - b)tan "B"/(2) = (a + b - c)tan "C"/(2)`.
Find the principal value of the following: tan- 1( - √3)
Prove the following:
`sin^-1(-1/2) + cos^-1(-sqrt(3)/2) = cos^-1(-1/2)`
sin−1x − cos−1x = `pi/6`, 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)`
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)]`
Find the principal value of `cos^-1 sqrt(3)/2`
Find the principal value of cosec–1(– 1)
Find the principal value of `tan^-1 (sqrt(3))`
`sin^-1x + sin^-1 1/x + cos^-1x + cos^-1 1/x` = ______
In ΔABC, tan`A/2 = 5/6` and tan`C/2 = 2/5`, then ______
The principle solutions of equation tan θ = -1 are ______
sin[3 sin-1 (0.4)] = ______.
If `sin^-1x + cos^-1y = (3pi)/10,` then `cos^-1x + sin^-1y =` ______
The principal value of `sin^-1 (sin (3pi)/4)` is ______.
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then θ = ______
If `tan^-1x + tan^-1y = (4pi)/5`, then `cot^-1x + cot^-1y` equals ______.
`(sin^-1(-1/2) + tan^-1(-1/sqrt(3)))/(sec^-1 (-2/sqrt(3)) + cos^-1(1/sqrt(2))` = ______.
The value of `sin^-1(cos (53pi)/5)` is ______
`cos^-1 4/5 + tan^-1 3/5` = ______.
Solve for x `tan^-1((1 - x)/(1 + x)) = 1/2 tan^-1x, x > 0`
Show that `2tan^-1 (-3) = (-pi)/2 + tan^-1 ((-4)/3)`
Show that `cos(2tan^-1 1/7) = sin(4tan^-1 1/3)`
All trigonometric functions have inverse over their respective domains.
If `"x + y" = "x"/4` then (1+ tanx)(1 + tany) is equal to ____________.
`("cos" 8° - "sin" 8°)/("cos" 8° + "sin" 8°)` is equal to ____________.
If `"sin"^-1("x"^2 - 7"x" + 12) = "n"pi, AA "n" in "I"`, then x = ____________.
If tan-1 3 + tan-1 x = tan-1 8, then x = ____________.
If tan-1 (x – 1) + tan-1 x + tan-1 (x + 1) = tan-1 3x, then the values of x are ____________.
The range of sin-1 x + cos-1 x + tan-1 x is ____________.
3 tan-1 a is equal to ____________.
If `"x" in (- pi/2, pi/2), "then the value of tan"^-1 ("tan x"/4) + "tan"^-1 ((3 "sin" 2 "x")/(5 + 3 "cos" 2 "x"))` is ____________.
`"tan"^-1 sqrt3 - "sec"^-1 (-2)` 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
The number of solutions of sin–1x + sin–1(1 – x) = cos–1x is
What will be the principal value of `sin^-1(-1/2)`?
Find the principal value of `tan^-1 (sqrt(3))`
Values of tan–1 – sec–1(–2) is equal to
`sin(tan^-1x), |x| < 1` is equal to
`2tan^-1 (cos x) = tan^-1 (2"cosec" x)`, then 'x' will be equal to
`lim_(n→∞)tan{sum_(r = 1)^n tan^-1(1/(1 + r + r^2))}` is equal to ______.
If sin–1a + sin–1b + sin–1c = π, then find the value of `asqrt(1 - a^2) + bsqrt(1 - b^2) + csqrt(1 - c^2)`.
sin [cot–1 (cos (tan–1 x))] = ______.
If y = `tan^-1 (sqrt(1 + x^2) - sqrt(1 - x^2))/(sqrt(1 + x^2) + sqrt(1 - x^2))`, then `dy/dx` is equal to ______.
Find the value of `cos(x/2)`, if tan x = `5/12` and x lies in third quadrant.
Solve for x:
5tan–1x + 3cot–1x = 2π
