Question

Which function is an odd function?

Answer 1

I think the answer is the first one, because it needs to be negative.  <span />

Answer 2

For any trigonometric point P(x,y)

x always represents cos

$x=cos\theta$

y always represents sin.

$y=sin\theta$

Now if we drop a perpendicular from P(x,y) to a point Q which is a refelction of P across x axis, we get Q(x, -y) for the same angle.

The angle shall be $-\theta$

So now

$cos(\-theta)=x and sin(-\theta) = -y$ ...(1)

But $x=cos\theta and y = sin\theta$

Statement (1) becomes

$cos(-\theta) = cos\theta and sin(-\theta) = -sin\theta$

So the value of cos does not change, but the value of sin changes.

Cos is even & sin is odd.

And so sec is even and cosec is odd.

So $f(x) = csc(\frac{-3\pi}{2})$ shall be an odd function.

Option B) is the right answer