Advertisements
Advertisements
प्रश्न
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Advertisements
उत्तर
Let `cos^(-1) (a/b) = 0`
Then `cos theta = a/b`
L.H.S:
`tan {pi/4 + 1/4 cos^(-1) a/b} + tan (pi/4 - 1/2cos^(-1) a/b)`
= `tan (pi/4 + theta/2) + tan (pi/4 - theta/2)`
`= (1 + tan theta/2)/(1 - tan theta/2) + (1 - tan theta/2)/(1 + tan theta/2)`
`((1 + tan theta/2)^2 + (1 - tan theta/2)^2)/(1 - tan^2 theta/2)`
= `2((1 + tan^2 theta/2)/(1 - tan^2 theta/2))`
= `2/(cos theta)` `[∵ cos 2 theta = (1 - tan^2 theta)/(1 + tan^2 theta)]`
= `(2b)/a`
=RHS
LHS = RHS
Hence Proved
संबंधित प्रश्न
Prove that `2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)=pi/4`
Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Find the value of the given expression.
`tan^(-1) (tan (3pi)/4)`
Solve the following equation:
2 tan−1 (cos x) = tan−1 (2 cosec x)
Solve the following equation for x: `cos (tan^(-1) x) = sin (cot^(-1) 3/4)`
Solve for x : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .
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
`sin^-1 (cos pi)`
Find the value of the expression in terms of x, with the help of a reference triangle
sin (cos–1(1 – x))
Solve: `sin^-1 5/x + sin^-1 12/x = pi/2`
Evaluate tan (tan–1(– 4)).
Prove that `2sin^-1 3/5 - tan^-1 17/31 = pi/4`
Solve the equation `sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2`
Show that `tan(1/2 sin^-1 3/4) = (4 - sqrt(7))/3` and justify why the other value `(4 + sqrt(7))/3` is ignored?
If `sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2))`. where a, x ∈ ] 0, 1, then the value of x is ______.
Solve for x : `"sin"^-1 2 "x" + sin^-1 3"x" = pi/3`
If tan-1 2x + tan-1 3x = `pi/4,` then x is ____________.
If `"tan"^-1 (("x" - 1)/("x" + 2)) + "tan"^-1 (("x" + 1)/("x" + 2)) = pi/4,` then x is equal to ____________.
Solve for x : `"sin"^-1 2"x" + "sin"^-1 3"x" = pi/3`
If `6"sin"^-1 ("x"^2 - 6"x" + 8.5) = pi,` then x is equal to ____________.
`"cos"^-1 1/2 + 2 "sin"^-1 1/2` is equal to ____________.
`"cos"^-1 (1/2)`
If `3 "sin"^-1 ((2"x")/(1 + "x"^2)) - 4 "cos"^-1 ((1 - "x"^2)/(1 + "x"^2)) + 2 "tan"^-1 ((2"x")/(1 - "x"^2)) = pi/3` then x is equal to ____________.
What is the value of cos (sec–1x + cosec–1x), |x| ≥ 1
`sin^-1(1 - x) - 2sin^-1 x = pi/2`, tan 'x' is equal to
Find the value of `tan^-1 [2 cos (2 sin^-1 1/2)] + tan^-1 1`.
If sin–1x + sin–1y + sin–1z = π, show that `x^2 - y^2 - z^2 + 2yzsqrt(1 - x^2) = 0`
