Advertisements
Advertisements
Question
Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`
Solution
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
RELATED QUESTIONS
If `sin (sin^(−1)(1/5)+cos^(−1) x)=1`, then find the value of x.
Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `
Prove that `2tan^(-1)(1/5)+sec^(-1)((5sqrt2)/7)+2tan^(-1)(1/8)=pi/4`
Prove the following:
`3sin^(-1) x = sin^(-1)(3x - 4x^3), x in [-1/2, 1/2]`
Find the value of the following:
`tan^-1 [2 cos (2 sin^-1 1/2)]`
Prove that:
`tan^(-1) 63/16 = sin^(-1) 5/13 + cos^(-1) 3/5`
Prove that
\[2 \tan^{- 1} \left( \frac{1}{5} \right) + \sec^{- 1} \left( \frac{5\sqrt{2}}{7} \right) + 2 \tan^{- 1} \left( \frac{1}{8} \right) = \frac{\pi}{4}\] .
Solve for x : \[\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)\] .
Find the value of `sin^-1[cos(sin^-1 (sqrt(3)/2))]`
Prove that `tan^-1 2/11 + tan^-1 7/24 = tan^-1 1/2`
If tan–1x + tan–1y + tan–1z = π, show that x + y + z = xyz
Simplify: `tan^-1 x/y - tan^-1 (x - y)/(x + y)`
Choose the correct alternative:
If `sin^-1x + cot^-1 (1/2) = pi/2`, then x is equal to
Prove that cot–17 + cot–18 + cot–118 = cot–13
If cos–1x > sin–1x, then ______.
If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.
The maximum value of sinx + cosx is ____________.
The value of sin (2tan-1 (0.75)) is equal to ____________.
If x = a sec θ, y = b tan θ, then `("d"^2"y")/("dx"^2)` at θ = `π/6` is:
`"sin"^-1 (1/sqrt2)`
`"tan"^-1 (sqrt3)`
`"sin"^-1 ((-1)/2)`
Solve for x : `{"x cos" ("cot"^-1 "x") + "sin" ("cot"^-1 "x")}^2` = `51/50
What is the value of cos (sec–1x + cosec–1x), |x| ≥ 1
Find the value of `cos^-1 (1/2) + 2sin^-1 (1/2) ->`:-
`sin^-1(1 - x) - 2sin^-1 x = pi/2`, tan 'x' is equal to
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
Write the following function in the simplest form:
`tan^-1 ((cos x - sin x)/(cos x + sin x)), (-pi)/4 < x < (3 pi)/4`
`"tan" ^-1 sqrt3 - "cot"^-1 (- sqrt3)` is equal to ______.
