Question

Can you check and see if I am doing this right

Answer 1

Given:

`\cos ^2\mleft((1)/(2)\cos ^(-1)\mleft(-(4)/(5)\mright)\mright)`

To find the exact value:

We know the formula,

`\cos ^2x=(1+\cos 2x)/(2)`

If we take,

`x=((1)/(2)\cos ^(-1)(-(4)/(5)))`

Then, the given problem can be rewritten as,

`\begin{gathered} \cos ^2((1)/(2)\cos ^(-1)(-(4)/(5)))=(1+\cos\lbrack2((1)/(2)\cos^(-1)(-(4)/(5)))\rbrack)/(2) \\ =(1+\cos \lbrack(2)/(2)\cos ^(-1)(-(4)/(5))\rbrack)/(2) \\ =(1+\cos \lbrack\cos ^(-1)(-(4)/(5))\rbrack)/(2) \end{gathered}`

Cancelling cos and its inverse we get

`\begin{gathered} =(1+(-(4)/(5)))/(2) \\ =(1-(4)/(5))/(2) \\ =((1)/(5))/(2) \\ =(1)/(10) \end{gathered}`

Hence, the exact value is,

`(1)/(10)`