Simplify the following inverse trigonomery function: \$\arc "cos" ("x") + \arc "cos"("y")\$.

The given inverse trigonometry function is

\$\arc "cos" ("x") + \arc "cos"("y")\$

Let’s denote the given function with a and b

\$"cos"^-1 ("x") = "a"\$ and \$ "cos"^-1("y") = "b"\$

From \$"cos"^-1 ("x") = "a"\$ we get , x = cos a

From \$"cos"^-1("y") = "b"\$ we get, y = cos b

The formula for invesre trigonommetry function is

\$"cos"("a" + "b")\$ = cos a . cos b - sin a . sin b

Rewrite the given function into formula

\$"cos"("a" + "b")\$ = cos a . cos b - sin a . sin b,\$("sin" "a" = \sqrt(1 - "cos"^2 ("a")))\$

Simplify the function

\$"cos"("a" + "b") = "cos" "a". "cos" "b" - \sqrt(1 - "cos"^2 "b") . \sqrt(1 - "cos"^2 "a")\$

So, here put cos a = x and cos b = y and then simpify the function

\$"cos"("a" + "b") = "xy" - \sqrt(1 - "x"^2) . \sqrt(1 - "y"^2)\$

\$"a" + "b" = "cos"^-1 ("xy" - \sqrt(1 - "x"^2) . \sqrt(1 - "y"^2))\$

So, the inverse trigonomery function: \$arc"cos"("x") + arc"cos"("y")\$ is \$"cos"^-1 ("xy" - \sqrt(1 - "x"^2) . \sqrt(1 - "y"^2))\$.

Correct simplified form, accurately reflecting the derived expression using standard notation for trigonometric identities

\$"cos"^-1 ("xy" - \sqrt(1 - "x"^2) . \sqrt(1 - "y"^2))\$.

It does not represent a valid simplification according to the trigonometric identities used

\$"cos"^-1("xy" + \sqrt(1 + "x"^2)). \sqrt(1 + "y"^2)\$

Option C does not represent a valid simplification according to the trigonometric identities used

\$"cos"^-1("xy" + \sqrt(1 + "x"^2)). \sqrt(1 - "y"^2)\$

It does not represent a valid simplification according to the trigonometric identities used

\$"cos"^-1("xy" - \sqrt(1 - "x"^2)). \sqrt(1 - "y"^2)\$

Option

A

Can you summarize what you’ve understood in the above steps?