Question

Prove csc (pi/2 - x) = sec x

Answer 1

Hello!

The correct answer is: D.
`cscx((\pi )/(2)-x)=(1)/(sin(\pi )/(2) *cosx-cos(\pi )/(2) *sinx)=secx`

Why?

Let's put a letter to each option in order to make the explanation easier:

First option will be A

Second option will be B

Third option will be C

Fourth option will be D

So,

First we need to know that:

`cscx=(1)/(sinx)`

and

`sin(x-y)=sinx*cosy-cosx*siny`

Therefore:

`secx=(1)/(sin((\pi )/(2)-x) )=(1)/(sin(\pi )/(2) *cosx-cos(\pi )/(2) *sinx)=secx`

`secx=(1)/(1*cosx-0 *sinx)=(1)/(cosx)=secx`

Then,

`secx=secx=cscx((\pi )/(2)-x)`

Have a nice day!

Answer 2

The answer is the last one ⇒
`csc((\pi )/(2)-x)=(1)/(sin(\pi )/(2)cosx-cos(\pi )/(2)sinx  )`

Explanation:

∵ sin(π/2 - x) = sinπ/2 cosx - cosπ/2 sinx

∵ sin(π/2) = 1 , ∵ cos(π/2) = 0

∴ sin(π/2 - x) = (1) × cosx - (0) sinx = cosx

∵ csc(π/2 - x) =
`(1)/(sin(\pi )/(2)-x )`

∴ csc(π/2 - x) = 1/cosx = secx