Advertisements
Advertisements
Question
Show that `cos(2tan^-1 1/7) = sin(4tan^-1 1/3)`
Advertisements
Solution
L.H.S. `cos(2tan^-1 1/7)`
= `cos[cos^-1 (1 - 1/49)/(1 + 1/49)]` .....`[because 2tan^-1x = cos^-1 (1 - x^2)/(1 + x^2)]`
= `cos[cos^-1 48/50]`
= `cos[cos^-1 24/25]`
= `24/25`
R.H.S `sin[4 tan^-1 1/3]`
= `sin[2tan^-1 ((2 xx 1/3)/(1 - 1/9))]` .....`[because 2tan^-1x = tan^-1 (2x)/(1 - x^2)]`
= `sin[2tan^-1 ((2/3)/(8/9))]`
= `sin[2tan^-1 3/4]`
= `sin[sin^-1 (2 xx 3/4)/(1 + 9/16)]` ......`[because 2tan^-1x = sin^-1 (2x)/(1 + x^2)]`
= `sin[sin^-1 24/25]`
⇒ `24/25`
L.H.S. = R.H.S.
Hence poved.
APPEARS IN
RELATED QUESTIONS
Find the principal value of the following:
`cos^(-1) (sqrt3/2)`
Find the principal value of the following:
`sec^(-1) (2/sqrt(3))`
`sin^-1{cos(sin^-1 sqrt3/2)}`
Evaluate the following:
`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`
Find the set of values of `cosec^-1(sqrt3/2)`
Evaluate the following:
`tan^-1(-1/sqrt3)+cot^-1(1/sqrt3)+tan^-1(sin(-pi/2))`
Solve for x:
`tan^-1 [(x-1),(x-2)] + tan^-1 [(x+1),(x+2)] = x/4`
In ΔABC, if a = 18, b = 24, c = 30 then find the values of cos `A/2`
In ΔABC prove that `(b + c - a) tan "A"/(2) = (c + a - b)tan "B"/(2) = (a + b - c)tan "C"/(2)`.
In ΔABC, prove the following:
`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`
Find the value of `cos^-1 (1/2) + tan^-1 (1/sqrt(3))`
If tan−1x + tan−1y + tan−1z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1
Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`
Find the principal value of the following:
`sin^-1 (- 1/2)`
Prove that:
`tan^-1 (4/3) + tan^-1 (1/7) = pi/4`
Solve `tan^-1 2x + tan^-1 3x = pi/4`
In ΔABC, tan`A/2 = 5/6` and tan`C/2 = 2/5`, then ______
If 2tan-1 (cos x) = tan-1 (cosec2 x), then x = ______.
`sin{tan^-1((1 - x^2)/(2x)) + cos^-1((1 - x^2)/(1 + x^2))}` is equal to ______
Prove that `cot(pi/4 - 2cot^-1 3)` = 7
All trigonometric functions have inverse over their respective domains.
If `"sin"^-1("x"^2 - 7"x" + 12) = "n"pi, AA "n" in "I"`, then x = ____________.
`"sin"^-1 (1 - "x") - 2 "sin"^-1 "x" = pi/2`
`"tan"(pi/4 + 1/2 "cos"^-1 "x") + "tan" (pi/4 - 1/2 "cos"^-1 "x") =` ____________.
`sin[π/3 - sin^-1 (-1/2)]` is equal to:
If `"sin"^-1("x"^2 - 7"x" + 12) = "n"pi, AA "n" in "I"`, then x = ____________.
`2"tan"^-1 ("cos x") = "tan"^-1 (2 "cosec x")`
If a = `(2sin theta)/(1 + costheta + sintheta)`, then `(1 + sintheta - costheta)/(1 + sintheta)` is
What is the value of `sin^-1(sin (3pi)/4)`?
What is the principal value of cosec–1(2).
Find the principal value of `tan^-1 (sqrt(3))`
If f(x) = x5 + 2x – 3, then (f–1)1 (–3) = ______.
Number of values of x which lie in [0, 2π] and satisfy the equation
`(cos x/4 - 2sinx) sinx + (1 + sin x/4 - 2cosx)cosx` = 0
If ax + b (sec (tan–1 x)) = c and ay + b (sec.(tan–1 y)) = c, then `(x + y)/(1 - xy)` = ______.
If tan–1 (2x) + tan–1 (3x) = `π/4`, then x = ______.
If 2 tan–1 (cosx) = tan–1 (2 cosec x), then sin x + cos x is equal to ______.
If cos–1 x > sin–1 x, then ______.
