Question

If x=1/3, then the value of cos(2cos⁻¹x+sin⁻¹x)=........,Select Proper option from the given options.
(a) -√8/9
(b) -√1/3
(c) √3/2
(d) 1/2

Answer 1

we have to find value of cos(2cos^-1x + sin^-1x) , where x = 1/3

Let , 2cos^-1x = A
cosA/2 = x
cosA = 2cos²A/2 - 1 [ from formula ]
= 2x² - 1 .......(i)

=> sinA =√{1 - (2x² - 1)²}
sinA =√(-4x⁴ +4x²) = 2x√(1 - x²) .......(ii)

similarly, sin^-1x = B
sinB = x ........(iii)
=> cosB = √(1 - x²) .........(iv)

now, cos(cos^-1x + sin^-1x) = cos(A + B)

= cosA.cosB - sinA.sinB [ from formula]

from eqs (i), (ii), (iii) and (iv)

=(2x² - 1).√(1 - x²) - 2x√(1 - x²).x

= - √(1 - x²)

now, put x = 1/3

= - √(1 - 1/3²)

= -√8/9

hence, option (a) is correct