Question

Write the expression as an equivalent algebraic expression involving only x.

Answer 1

The given expression is $\cos \left(2 \cos ^{-1} x\right)$

We need to determine the expression as an equivalent algebraic expression involving only x.

The expression $\cos \left(2 \cos ^{-1} x\right)$ has to be expressed as an algebraic expression in x.

Thus, we have;

$\cos \left(2 \cos ^{-1} x\right)$

Using the identity $cos \ (2x)=2cos^2x-1$ in the above expression, we get;

$\cos \left(2 \cos ^{-1} x\right)=2\ cos^2(cos^{-1}x)-1$

Simplifying, we get;

$\cos \left(2 \cos ^{-1} x\right)=2[cos(cos^{-1}x)]^2-1$

$\cos \left(2 \cos ^{-1} x\right)=2(x)^2-1$

$\cos \left(2 \cos ^{-1} x\right)=2x^2-1$

Hence, the expression $\cos \left(2 \cos ^{-1} x\right)$ can be written as $2x^2-1$