Advertisements
Advertisements
प्रश्न
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
Advertisements
उत्तर
LHS = `tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8)`
`= tan^-1 [(1/5 + 1/7)/(1 - 1/5 xx 1/7)] + tan^-1 [(1/3+1/8)/(1 - 1/3 xx 1/8)]`
`= tan^-1 ((7 + 5)/(35 - 1)) + tan^-1 ((8 + 3)/(24 - 1))`
`= tan^-1 (12/34) + tan^-1 (11/23)`
`= tan^-1 (6/17) + tan^-1 (11/23)`
`= tan^-1 [(6/17 + 11/23)/(1 - 6/17 xx 11/23)]`
`= tan^-1((138 + 187)/(391 - 66)) = tan^-1 (325/325)`
`= tan^-1 (1) = tan^-1 (tan pi/4)`
`= pi/4`
= RHS.
APPEARS IN
संबंधित प्रश्न
In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.
(A) `1/5`
(B) `sqrt(1/5)`
(C) `4/5`
(D) `2/5`
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
The principal solutions of cot x = -`sqrt3` are .................
In , ΔABC prove that
`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
In any Δ ABC, prove the following:
a2 sin (B - C) = (b2 - c2) sin A.
In Δ ABC, if a, b, c are in A.P., then show that cot `"A"/2, cot "B"/2, cot "C"/2` are also in A.P.
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.
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
State whether the following equation has a solution or not?
cos 2θ = `1/3`
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`
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 `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 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, if (a+ b - c)(a + b + c) = 3ab, then ______.
In a ΔABC, cot `(("A - B")/2)* tan (("A + B")/2)` is equal to
In a ΔABC, c2 sin 2B + b2 sin 2C = ?
With usual notations, if the angles A, B, C of a Δ ABC are in AP and b : c = `sqrt3 : sqrt2`.
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 ______
In Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.
If P(6, 10, 10), Q(1, 0, -5), R(6, -10, λ) are vertices of a triangle right angled at Q, then value of λ is ______.
In ΔABC, `(sin(B - C))/(sin(B + C))` = ______
In Δ ABC, with the usual notations, if `(tan "A"/2)(tan "B"/2) = 3/4` then a + b = ______.
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC 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 ______.
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
If in a triangle ABC, AB = 5 units, AB = 5 units, ∠B = `cos^-1 (3/5)` and radius of circumcircle of ΔABC is 5 units, then the area (in sq.units) of ΔABC is ______.
If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In any ΔABC, prove that:
(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c.
The perimeter of ΔABC is 20, ∠A = 60°, area of ΔABC = `10sqrt(3)`, then find the values of a, b, c.
In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.
