We're going to verify this by working on the left side. We'll be using the reciprocal identity and some algebra skills.
cscx(1 + cscx) = [tex] \frac{sinx + 1}{sin^2x} [/tex] Go ahead and write the left side by itself.
cscx (1 + cscx) Distribute the cscx into the parentheses (mulitply)
cscx + csc²x Use the reciprocal identity to change cscx into [tex] \frac{1}{sinx} [/tex]
[tex] \frac{1}{sinx} [/tex] + csc²x Use the same identity to change csc²x into [tex] \frac{1}{sin^2x} [/tex]
[tex] \frac{1}{sinx} [/tex] + [tex] \frac{1}{sin^2x} [/tex] Multiply [tex] \frac{1}{sinx} [/tex] by [tex] \frac{sinx}{sinx} [/tex] (it's the same as multiplying it by 1)
[tex] \frac{sinx}{sin^2x} [/tex] + [tex] \frac{1}{sin^2x} [/tex] Combine the numerators