Question

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

Answer 1

Answer:

The answer is the last one ⇒ $csc(\frac{\pi }{2}-x)=\frac{1}{sin\frac{\pi }{2}cosx-cos\frac{\pi }{2}sinx }$

Step-by-step 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) = $\frac{1}{sin\frac{\pi }{2}-x }$

∴ csc(π/2 - x) = 1/cosx = secx

Answer 2

Hello!

The correct answer is: D. $cscx(\frac{\pi }{2}-x)=\frac{1}{sin\frac{\pi }{2} *cosx-cos\frac{\pi }{2} *sinx}=secx$

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:

$cscx=\frac{1}{sinx}$

and

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

Therefore:

$secx=\frac{1}{sin(\frac{\pi }{2}-x) }=\frac{1}{sin\frac{\pi }{2} *cosx-cos\frac{\pi }{2} *sinx}=secx$

$secx=\frac{1}{1*cosx-0 *sinx}=\frac{1}{cosx}=secx$

Then,

$secx=secx=cscx(\frac{\pi }{2}-x)$

Have a nice day!