The value of Sin(2tan^-11/3)+cos(tan^-12√2) is?

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Solution

we have Sin(2tan^-11/3)+cos(tan^-12√2)

Sin(2tan^-11/3) =sin { sin^-1 [( 2*1/3/(1+(1/3)^2 )]}

(because 2tan^-1x=sin^-1[2x/1+x^2]

Sin(2tan^-11/3) = sin{sin^-1[(2/3)/(10/9)]}

= sin [sin^-1[3/5]] =3/5 (because sin(sin^-1x) =x )

cos(tan^-12√2) = cos [cos^-1(1/3)] =1/3
because cos [cos^-1(x)] =x

therefore

Sin(2tan^-11/3)+cos(tan^-12√2) = 3/5+1/3=9+5/15 = 14/15

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