Simplifying Expressions With Trigonometric Functions of Inverse Trigonometric Functions

We can simplify expressions with sines and cosines of inverse trigonometric functions using substitutions.
To find  
\[sin(2 sin^{-1}x)\]
  substitute  
\[\theta = sin^{-1}x\]
  then
\[sin(2 sin^{-1}x)=sin(2 \theta)=2 sin \theta cos \theta =2x \sqrt{1-x^2}\]

(
\[cos \theta = \sqrt{1-sin^2 \theta}=\sqrt{1-x^2}\]
)

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