Question

Express the following as an algebraic function of x.sin(sin-'(x) + cosos-'(x))

Answer 1

`2x^2-1`

Step-by-step explanation:

Given the below function;

`\sin (\sin ^(-1)(x)+\cos ^(-1)(x))`

Let;

`\begin{gathered} \sin ^(-1)\mleft(x\mright)=a \\ \cos ^(-1)\mleft(x\mright)=b \end{gathered}`

So we'll have;

`\begin{gathered} \sin a=\cos b=x \\ \therefore\cos a=\sin b=\pm\sqrt[]{1-x^2} \end{gathered}`

We can now rewrite the given expression as;

`\begin{gathered} \sin (a+b)=\sin a\cos b+\cos a\sin b \\ =x\cdot x+(\pm\sqrt[]{1-x^2})\cdot(\pm\sqrt[]{1-x^2}) \\ =x^2\pm(1-x^2) \\ So\text{ we'll have;} \\ \Rightarrow x^2+(1-x^2) \\ x^2+1-x^2=1 \\ Or \\ \Rightarrow x^2-(1-x^2) \\ x^2-1+x^2=2x^2-1 \end{gathered}`

We can see from the above that the algebraic function of x is 2x^2 - 1