Advertisements
Advertisements
प्रश्न
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
Advertisements
उत्तर
Let `tan^-1 sqrt"x"` = y
∴ tan y = `sqrt"x"`
∴ x = tan2y
Now,
RHS = `1/2 cos^-1 ((1 - "x")/(1 + "x"))`
`= 1/2 cos^-1 ((1 - tan^2 "y")/(1 + tan^2 "y"))`
`= 1/2 cos^-1 (cos 2"y")`
`= 1/2 (2"y") = "y"`
`= tan^-1 sqrt"x"`
= LHS.
APPEARS IN
संबंधित प्रश्न
In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`.
In any ΔABC, with usual notations, prove that b2 = c2 + a2 – 2ca cos B.
Find the Cartesian coordinates of the point whose polar coordinates are :
`(4, pi/2)`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(3/4, (3pi)/4)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(0, 1/2)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(3/2, (3√3)/2)`.
In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.
In any Δ ABC, prove the following:
a2 sin (B - C) = (b2 - c2) sin A.
In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
In Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
In Δ ABC, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
In Δ ABC, if a cos2 `"C"/2 + "c cos"^2 "A"/2 = "3b"/2`, then prove that a, b, c are in A.P.
Show that `2 sin^-1 (3/5) = tan^-1(24/7)`
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
In ΔABC, a = 3, b = 4 and sin A = `3/4`, find ∠B
In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled
In ∆ABC, prove that `sin ((A - B)/2) = ((a - b)/c) cos C/2`
If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a2, b2, c2 are in A.P.
In ΔABC, prove that `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0
In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
With usual notations, if the angles A, B, C of a Δ ABC are in AP and b : c = `sqrt3 : sqrt2`.
In a triangle ABC, If `(sin "A" - sin "C")/(cos "C" - cos "A")` = cot B, then A, B, C are in ________.
If in a right-angled triangle ABC, the hypotenuse AB = p, then `overline"AB".overline" AC" + overline"BC".overline" BA" + overline" CA".overline"CB"` is equal to ______
If one side of a triangle is double the other and the angles opposite to these sides differ by 60°, then the triangle is ______
If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______
In ΔABC, a = 7cm, b = 3cm and c = 8 cm, then angle A is ______
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
In a ΔABC, if `("b" + "c")/11 = ("c" + "a")/12 = ("a" + "b")/13`, then cos C = ______.
Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.
Let ABC be a triangle such that ∠A = 45°, ∠B = 75° then `"a" + "c"sqrt(2)` is equal to ______. (in usual notation)
In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.
In ΔABC, with usual notations, if a, b, c are in A.P. Then `a cos^2 (C/2) + c cos^2(A/2)` = ______.
In ΔABC, `(a - b)^2 cos^2 C/2 + (a + b)^2 sin^2 C/2` is equal to ______.
In a triangle ABC, with usual notations, if \[\frac{2\cos\mathrm{A}}{\mathrm{a}}+\frac{\cos\mathrm{B}}{\mathrm{b}}+\frac{2\cos\mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ac}}\]then ∠A =
In a triangle ABC with usual notations, if a,b, and c are in arithmetic progression, then, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
