Advertisements
Advertisements
प्रश्न
Show that `2tan^-1 {tan alpha/2 * tan(pi/4 - beta/2)} = tan^-1 (sin alpha cos beta)/(cosalpha + sinbeta)`
उत्तर
L.H.S. = `tan^-1 (2tan alpha/2 * tan (pi/4 - beta/2))/(1 - tan^2 alpha/2 tan^2 (pi/4 - beta/2))` ......`("since" 2 tan^-1x = tan^-1 (2x)/(1 - x^2))`
= `tan^-1 (2tan alpha/2 (1 - tan beta/2)/(1 + tan beta/2))/(1 - tan^2 alpha/2 ((1 - tan beta/2)/(1 + tan beta/2))^2)`
= `tan^-1 (2tan alpha/2 (1 - tan^2 beta/2))/((1 + tan beta/2)^2 - tan^2 alpha/2 (1 - tan beta/2)^2)`
= `tan^-1 (2tan alpha/2 (1 - tan^2 beta/2))/((1 + tan^2 beta/2)(1 - tan^2 alpha/2) + 2 beta/2 (1 + tan^2 alpha/2))`
= `tan^-1 ((2tan alpha/2)/(1 + tan^2 alpha/2) - (1 - tan^2 beta/2)/(1 + tan^2 beta/2))/((1 - tan^2 alpha/2)/(1 + tan^2 alpha/2) + (2tan beta/2)/(1 + tan^ beta/2))`
= `tan^-1 ((sin alpha cos beta)/(cos alpha + sin beta))`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`
Find the value of `cot(tan^(-1) a + cot^(-1) a)`
Solve `tan^(-1) - tan^(-1) (x - y)/(x+y)` is equal to
(A) `pi/2`
(B). `pi/3`
(C) `pi/4`
(D) `(-3pi)/4`
Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`
Solve for x : `tan^-1 ((2-"x")/(2+"x")) = (1)/(2)tan^-1 ("x")/(2), "x">0.`
Find the value of `cot[sin^-1 3/5 + sin^-1 4/5]`
Prove that `tan^-1 2/11 + tan^-1 7/24 = tan^-1 1/2`
Prove that `sin^-1 3/5 - cos^-1 12/13 = sin^-1 16/65`
Prove that `tan^-1x + tan^-1 (2x)/(1 - x^2) = tan^-1 (3x - x^3)/(1 - 3x^2), |x| < 1/sqrt(3)`
Solve: `2tan^-1 (cos x) = tan^-1 (2"cosec" x)`
Choose the correct alternative:
If `cot^-1(sqrt(sin alpha)) + tan^-1(sqrt(sin alpha))` = u, then cos 2u is equal to
Evaluate `tan^-1(sin((-pi)/2))`.
Evaluate tan (tan–1(– 4)).
Solve the equation `sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2`
If α ≤ 2 sin–1x + cos–1x ≤ β, then ______.
Prove that `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/((1 + x^2) - sqrt(1 - x^2))) = pi/2 + 1/2 cos^-1x^2`
Prove that `sin^-1 8/17 + sin^-1 3/5 = sin^-1 7/85`
If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.
If cos–1x > sin–1x, then ______.
The maximum value of sinx + cosx is ____________.
If `"cot"^-1 (sqrt"cos" alpha) - "tan"^-1 (sqrt"cos" alpha) = "x",` the sinx is equal to ____________.
The value of sin (2tan-1 (0.75)) is equal to ____________.
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
`"tan"^-1 1/3 + "tan"^-1 1/5 + "tan"^-1 1/7 = "tan"^-1 1/8 =` ____________.
`"sin"^-1 (1 - "x") - 2 "sin"^-1 "x" = pi/2`
Solve for x : `{"x cos" ("cot"^-1 "x") + "sin" ("cot"^-1 "x")}^2` = `51/50
Find the value of `tan^-1 [2 cos (2 sin^-1 1/2)] + tan^-1 1`.
Solve for x: `sin^-1(x/2) + cos^-1x = π/6`
