Question

The solution set of the equation cos^-1x -sin^-1x =sin^-1(1-x) is ;

(a) [-1, 1]

(b) [0, 1/2]

(c) [-1, 0]

(d) none of these

Answer 1

cos^-1x -sin^-1x =sin^-1(1-x)

=>cos^-1x -(π/2-cos^-1x) =sin^-1(1-x) ;

=>2cos^-1x =π/2+sin^-1(1-x) ;

=>cos(2cos^-1x) =cos(π/2+sin^-1(1-x) );

=>2[coscos^-1x]²-1 =-sinsin^-1(1-x)

=> 2x²-1=-1+x

=>2x²-x=0

=> x(2x-1)=0

So x=0 and x=1/2

Hence option (b) is correct