Question

Sin^2x/cos^2 + sinxcscx = sec^2x Identify it, please explain so I know how you did it, thanks

Answer 1

to the risk of sounding redundant.

$\bf \stackrel{\textit{Pythagorean Identities}}{sin^2(\theta)+cos^2(\theta)=1}\quad \qquad csc(\theta )=\cfrac{1}{sin(\theta )}\qquad \qquad sec(\theta )=\cfrac{1}{cos(\theta )} \\\\ -------------------------------\\\\ \cfrac{sin^2(x)}{cos^2(x)}+sin(x)csc(x)=sec^2(x)\\\\ -------------------------------$

$\bf \cfrac{sin^2(x)}{cos^2(x)}+\underline{sin(x)}\cdot \cfrac{1}{\underline{sin(x)}}\implies \cfrac{sin^2(x)}{cos^2(x)}+1\implies \cfrac{sin^2(x)+cos^2(x)}{cos^2(x)} \\\\\\ \cfrac{1}{cos^2(x)}\implies sec^2(x)$

Answer 2

let s=sin(x), c=cos(x)

s^2/c^2+s*csc(x)

=s^2/c^2+s/s (csc(x) = 1/sin(x), by definition of csc(x) )

=s^2/c^2+1

=s^2/c^2 + c^2/c^2

=(s^2+c^2)/c^2 (s^2+c^2=1, an identity)

=1/c^2

= sec^2(x) (by definition of sec(x) = 1/cos(x) )