Advertisements
Advertisements
Question
Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`
Advertisements
Solution
Consider the left hand side
L.H.S = `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))`
We know that,
`tan^(-1)(A)-tan^(-1)(B)= tan^(-1)((A-B)/(1+AB))`
Thus, L.H.S = `tan^(-1)(((6x-8x^3)/(1-12x^2)-(4x)/(1-4x^2))/(1+((6x-8x^3)/(1-12x^2))((4x)/(1-4x^2))))`
`=tan^(-1)(((6x-8x^3)(1-4x^2)-4x(1-12x^2))/(((1-12x^2)(1-4x^2))/(1+(4x(6x-8x^3))/((1-12x^2)(1-4x^2)))))`
`=tan^(-1)((((6x-8x^3)(1-4x^2)-4x(1-12x^2))/((1-12x^2)(1-4x^2)))/(((1-12x^2)(1-4x^2)+4x(6x-8x^3))/((1-12x^2)(1-4x^2))))`
`=tan^(-1)(((6x-8x^3)(1-4x^2)-4x(1-12x^2))/((1-12x^2)(1-4x^2)+4x(6x-8x^3)))`
`=tan^(-1)((6x-24x^3-8x^3+32x^5-4x+48x^3)/(1-4x^2-12x^2+48x^4+24x^2-32x^4))`
`=tan^(-1)((32x^5+16x^3+2x)/(16x^4+8x^2+1))`
`=tan^(-1)((2x(16x^4+8x^2+1))/(16x^4+8x^2+1))`
= tan-12x
Thus, L.H.S=R.H.S
APPEARS IN
RELATED QUESTIONS
Prove that:
`tan^(-1)""1/5+tan^(-1)""1/7+tan^(-1)""1/3+tan^(-1)""1/8=pi/4`
If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.
Prove the following:
3 sin−1 x = sin−1 (3x − 4x3), `x ∈ [-1/2, 1/2]`
Prove `tan^(-1) 2/11 + tan^(-1) 7/24 = tan^(-1) 1/2`
Prove `2 tan^(-1) 1/2 + tan^(-1) 1/7 = tan^(-1) 31/17`
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) 1/(sqrt(x^2 - 1)), |x| > 1`
Prove that:
`tan^(-1) 63/16 = sin^(-1) 5/13 + cos^(-1) 3/5`
sin (tan–1 x), |x| < 1 is equal to ______.
sin–1 (1 – x) – 2 sin–1 x = `pi/2`, then x is equal to ______.
If y = `(x sin^-1 x)/sqrt(1 -x^2)`, prove that: `(1 - x^2)dy/dx = x + y/x`
Find: ∫ sin x · log cos x dx
Find the value, if it exists. If not, give the reason for non-existence
`sin^-1 [sin 5]`
Simplify: `tan^-1 x/y - tan^-1 (x - y)/(x + y)`
Evaluate `cos[cos^-1 ((-sqrt(3))/2) + pi/6]`
If a1, a2, a3,...,an is an arithmetic progression with common difference d, then evaluate the following expression.
`tan[tan^-1("d"/(1 + "a"_1 "a"_2)) + tan^-1("d"/(21 + "a"_2 "a"_3)) + tan^-1("d"/(1 + "a"_3 "a"_4)) + ... + tan^-1("d"/(1 + "a"_("n" - 1) "a""n"))]`
The maximum value of sinx + cosx is ____________.
If `"tan"^-1 ("cot" theta) = 2theta, "then" theta` is equal to ____________.
`"sin" {2 "cos"^-1 ((-3)/5)}` is equal to ____________.
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
If tan-1 2x + tan-1 3x = `pi/4,` then x is ____________.
sin (tan−1 x), where |x| < 1, is equal to:
`"tan" (pi/4 + 1/2 "cos"^-1 "x") + "tan" (pi/4 - 1/2 "cos"^-1 "x") =` ____________.
`"cos" (2 "tan"^-1 1/7) - "sin" (4 "sin"^-1 1/3) =` ____________.
If `"sin"^-1 (1 - "x") - 2 "sin"^-1 ("x") = pi/2,` then x is equal to ____________.
`sin^-1(1 - x) - 2sin^-1 x = pi/2`, tan 'x' is equal to
Find the value of `sin^-1 [sin((13π)/7)]`
`tan(2tan^-1 1/5 + sec^-1 sqrt(5)/2 + 2tan^-1 1/8)` is equal to ______.
Solve for x: `sin^-1(x/2) + cos^-1x = π/6`
