Simplify the expression below cos sin1 x tan1 xSolutioncos

Simplify the expression below: cos (sin^-1 x - tan^-1 x)

Solution

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

applying cos (a-b) formula

cos (a-b) = cos a cos b + sin a sin b

a = sin^-1 x , b = tan^-1 x

sin a = x ,tan b = x

cos a = sqrt ( 1-x^2 )

cos b = 1 / sqrt ( 1+x^2)

sin b = x / sqrt (1+x^2)

plugging the values in the formula

cos (a-b) = cos a cos b + sin a sin b

cos (a - b ) = sqrt (1-x^2) * 1 / sqrt ( 1+x^2) + x * x / sqrt (1+x^2)

cos ( a- b ) = sqrt ( 1-x^2) / sqrt (1+x^2) + x^2 / sqrt (1+x^2)

cos (a - b ) = { sqrt ( 1- x^2) + x^2 } / sqrt ( 1+x^2)

therefore,

cos ( sin^-1 x - tan^-1 x ) = { sqrt ( 1- x^2) + x^2 } / sqrt ( 1+x^2)