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`