$csc(x) = frac{1}{sin(x)}$

$sec(x) = frac{1}{cos(x)}$

$cot(x) = frac{1}{ an(x)} = frac{cos(x)}{sin(x)}$

Tangent: Defined everywhere except where $cos(x) = 0$ | Cotangent: Defined everywhere except where $sin(x) = 0$

Secant: $(-infty, -1] cup [1, infty)$ | Tangent: $(-infty, infty)$

sin(x): Period is $2pi$ | csc(x): Period is $2pi$

cos(x): Period is $2pi$ | sec(x): Period is $2pi$

tan(x): Period is $pi$ | cot(x): Period is $pi$

Cosecant is the reciprocal of sine. When sine is zero, cosecant is undefined (vertical asymptote). When sine is at its max/min, cosecant is at its min/max.

Secant is the reciprocal of cosine. When cosine is zero, secant is undefined (vertical asymptote). When cosine is at its max/min, secant is at its min/max.

Cotangent is the reciprocal of tangent. When tangent is zero or undefined, cotangent is undefined (vertical asymptote). Cotangent is decreasing while tangent is increasing.

All real numbers except $x = npi$, where n is an integer. (Where $sin(x) = 0$)

All real numbers except $x = frac{(2n+1)pi}{2}$, where n is an integer. (Where $cos(x) = 0$)

All real numbers except $x = npi$, where n is an integer. (Where $sin(x) = 0$)

$(-infty, -1] cup [1, infty)$

$(-infty, -1] cup [1, infty)$

$(-infty, infty)$

At $x = npi$, where n is an integer, because $sin(x) = 0$ there.

At $x = frac{(2n+1)pi}{2}$, where n is an integer, because $cos(x) = 0$ there.

At $x = npi$, where n is an integer, because $sin(x) = 0$ there.