Question

How do you verify `sec^2(x)+csc^2(x)=(sec(x)csc(x))^2`?

Answer 1

Use the inverse forms of sec and csc; apply rule `sin^2+cos^2 = 1` and perform standard algebraic operations.

Explanation:

`sec^2(x)+csc^2(x)`
`color(white)("XXXX")` `=1/(cos^2(x)) + 1/(sin^2(x))`

`color(white)("XXXX")` `=(sin^2(x)+cos^2(x))/(cos^2(x)*sin^2(x))`

`color(white)("XXXX")` `=1/(cos^2(x)*sin^2(x))`

`color(white)("XXXX")` `=1/cos^2(x)*1/sin^2(x)`

`color(white)("XXXX")` `= sec^2(x) * csc^2(x)`

`color(white)("XXXX")` `= (sec(x)csc(x))^2`