To complete the identity, we need these fundamental identities:
[tex]1)\displaystyle{sec(x)=\frac{1}{cos(x)}[/tex]
[tex]2) cos(x-y)=cos(x)cos(y)+sin(x)sin(y)[/tex]
[tex]\displaystyle{csc(x)= \frac{1}{sin(x)} [/tex]
Thus, by identity 1 we have:
[tex]\displaystyle{ sec( \frac{ \pi }{2}-\theta )= \frac{1}{cos(\frac{ \pi }{2}-\theta)} [/tex]
by identity :
[tex]\displaystyle{cos(\frac{ \pi }{2}-\theta)=cos(\frac{ \pi }{2})cos(\theta)+sin(\frac{ \pi }{2})sin(\theta)[/tex]
recall the values :
[tex]\displaystyle{ sin(\frac{ \pi }{2})^R=sin(90^o)=1\\\\[/tex]
[tex]\displaystyle{ cos(\frac{ \pi }{2})^R=cos(90^o)=0[/tex],
so:
[tex]cos(\frac{ \pi }{2})cos(\theta)+sin(\frac{ \pi }{2})sin(\theta)=0+sin(\theta)=sin(\theta)[/tex]
Putting all these together, we have:
[tex]\displaystyle{ sec( \frac{ \pi }{2}-\theta )= \frac{1}{cos(\frac{ \pi }{2}-\theta)}= \frac{1}{cos(\frac{ \pi }{2})cos(\theta)+sin(\frac{ \pi }{2})sin(\theta)}= \frac{1}{sin(\theta)}} [/tex]
which is equal to [tex]csc(\theta)[/tex], by identity 3
Answer: D
