Question

Can someone help me? Will fan and medal
tan(sin^-1(1/3)

Answer 1

Let\[y=\sin^{-1}(\frac{1}{3})\]\[\sin(y)=\frac{1}{3}\]\[\cos(y)=\sqrt{1-(\frac{1}{3})^2}\]\[\cos(y)=\sqrt{1-\frac{1}{9}}\]\[\cos(y)=\sqrt{\frac{8}{9}}\]\[\cos(y)=\frac{2\sqrt{2}}{3}\]\[\tan(y)=\frac{\sin(y)}{\cos(y)}=\frac{1}{3} \times \frac{3}{2\sqrt{2}}=\frac{1}{2\sqrt{2}}\]\[y=\tan^{-1}\frac{1}{2\sqrt{2}}\]\[\implies \sin^{-1}\frac{1}{3}=\tan^{-1}\frac{1}{2\sqrt{2}}\]\[\implies \tan(\sin^{-1}\frac{1}{3})=\tan(\tan^{-1}\frac{1}{2\sqrt{2}})=\frac{1}{2\sqrt{2}}\]
General Idea is to convert your inverse function to the corresponding trigonometric function so they cancel out. This is easily done by substituting a variable for the inverse function(step1), calculating the corresponding trigonometric function (step 2) then converting it to the required trigonometric function (step 3 to 7)) and finally converting that trigonometric function back to inverse (step 8 and 9) and sub back so it cancels and gives u answer (step 10)