Question

How do you verify `sin^2xsec^2x+sin^2xcsc^2x=sec^2x`?

Answer 1

See below for verification with the assumption that `sin(x)!=0`
The given equation is not true if `sin(x)=0`

Explanation:

`sin^2(x)sec^2(x)+sin^2(x)csc^2(x)`

`color(white)("XXX")=sin^2(x)[sec^2(x)+csc^2(x)]`

`color(white)("XXX")=sin^2(x)[1/cos^2(x)+1/(sin^2(x))]`

`color(white)("XXX")=sin^2(x)[(sin^2(x)+cos^2(x))/(cos^2(x)sin^2(x))]`

`color(white)("XXX")=sin^2(x)[1/(cos^2(x)sin^2(x))]`

`color(white)("XXX")=1/(cos^2(x))`

`color(white)("XXX")=sec^2(x)`