Question

10 points!!! please help :(

Answer 1

To complete the identity, we need these fundamental identities:


`1)\displaystyle{sec(x)=(1)/(cos(x))`


`2) cos(x-y)=cos(x)cos(y)+sin(x)sin(y)`


`\displaystyle{csc(x)= (1)/(sin(x))`


Thus, by identity 1 we have:


`\displaystyle{ sec( ( \pi )/(2)-\theta )= (1)/(cos(( \pi )/(2)-\theta))`

by identity :


`\displaystyle{cos(( \pi )/(2)-\theta)=cos(( \pi )/(2))cos(\theta)+sin(( \pi )/(2))sin(\theta)`

recall the values :


`\displaystyle{ sin(( \pi )/(2))^R=sin(90^o)=1\\\\`


`\displaystyle{ cos(( \pi )/(2))^R=cos(90^o)=0`,


so:


`cos(( \pi )/(2))cos(\theta)+sin(( \pi )/(2))sin(\theta)=0+sin(\theta)=sin(\theta)`


Putting all these together, we have:



`\displaystyle{ sec( ( \pi )/(2)-\theta )= (1)/(cos(( \pi )/(2)-\theta))= (1)/(cos(( \pi )/(2))cos(\theta)+sin(( \pi )/(2))sin(\theta))= (1)/(sin(\theta))}`

which is equal to
`csc(\theta)`, by identity 3


Answer: D