Sum of Tangent and Cotangent

Theorem

$\tan x + \cot x = \sec x \csc x$

Proof

\(\ds \tan x + \cot x\)

\(=\)

\(\ds \frac {\sin x} {\cos x} + \cot x\)

Tangent is Sine divided by Cosine

\(\ds \)

\(=\)

\(\ds \frac {\sin x} {\cos x} + \frac {\cos x} {\sin x}\)

Cotangent is Cosine divided by Sine

\(\ds \)

\(=\)

\(\ds \frac {\sin^2 x + \cos^2x} {\cos x \sin x}\)

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos x \sin x}\)

Sum of Squares of Sine and Cosine

\(\ds \)

\(=\)

\(\ds \sec x \frac 1 {\sin x}\)

Secant is Reciprocal of Cosine

\(\ds \)

\(=\)

\(\ds \sec x \csc x\)

Cosecant is Reciprocal of Sine

$\blacksquare$