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
If `sin (sin^(−1) 1/5+cos^(−1) x)=1`, then find the value of x.
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Prove the following:
3cos–1x = cos–1 (4x3 – 3x), `x ∈ [1/2, 1]`
Write the following function in the simplest form:
`tan^(-1) (sqrt(1 + x^2) - 1)/x, x ≠ 0`
Write the function in the simplest form: `tan^(-1) ((cos x - sin x)/(cos x + sin x)) `,` 0 < x < pi`
Find the value of the following:
`tan^-1 [2 cos (2 sin^-1 1/2)]`
if `sin(sin^(-1) 1/5 + cos^(-1) x) = 1` then find the value of x
Find the value of the given expression.
`sin^(-1) (sin (2pi)/3)`
Prove `(9pi)/8 - 9/4 sin^(-1) 1/3 = 9/4 sin^(-1) (2sqrt2)/3`
sin (tan–1 x), |x| < 1 is equal to ______.
If cos-1 x + cos -1 y + cos -1 z = π , prove that x2 + y2 + z2 + 2xyz = 1.
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
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)`.
Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1 ("x")/(2), "x">0.`
If tan–1x + tan–1y + tan–1z = π, show that x + y + z = xyz
Find the number of solutions of the equation `tan^-1 (x - 1) + tan^-1x + tan^-1(x + 1) = tan^-1(3x)`
If `tan^-1x = pi/10` for some x ∈ R, then the value of cot–1x is ______.
The maximum value of sinx + cosx is ____________.
If `"sec" theta = "x" + 1/(4 "x"), "x" in "R, x" ne 0,`then the value of `"sec" theta + "tan" theta` is ____________.
`"sin" {2 "cos"^-1 ((-3)/5)}` is equal to ____________.
If x = a sec θ, y = b tan θ, then `("d"^2"y")/("dx"^2)` at θ = `π/6` is:
Simplest form of `tan^-1 ((sqrt(1 + cos "x") + sqrt(1 - cos "x"))/(sqrt(1 + cos "x") - sqrt(1 - cos "x")))`, `π < "x" < (3π)/2` is:
Solve for x : `"sin"^-1 2"x" + "sin"^-1 3"x" = pi/3`
The value of `"tan"^-1 (3/4) + "tan"^-1 (1/7)` is ____________.
`"sin"^-1 (1 - "x") - 2 "sin"^-1 "x" = pi/2`
`"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 ____________.
The Government of India is planning to fix a hoarding board at the face of a building on the road of a busy market for awareness on COVID-19 protocol. Ram, Robert and Rahim are the three engineers who are working on this project. “A” is considered to be a person viewing the hoarding board 20 metres away from the building, standing at the edge of a pathway nearby. Ram, Robert and Rahim suggested to the firm to place the hoarding board at three different locations namely C, D and E. “C” is at the height of 10 metres from the ground level. For viewer A, the angle of elevation of “D” is double the angle of elevation of “C” The angle of elevation of “E” is triple the angle of elevation of “C” for the same viewer. Look at the figure given and based on the above information answer the following:
Measure of ∠CAB = ________.
What is the simplest form of `tan^-1 sqrt(1 - x^2 - 1)/x, x ≠ 0`
`cos^(−1)(1/2) + sin^(−1)(1) + tan^(−1) 1/sqrt3` is equal to ______.
