Tangent times Tangent plus Cotangent/Proof 1

Theorem

$\tan x \paren {\tan x + \cot x} = \sec^2 x$

Proof

\(\ds \tan x \left({\tan x + \cot x}\right)\)

\(=\)

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

Sum of Tangent and Cotangent

\(\ds \)

\(=\)

\(\ds \frac {\sin x} {\cos x} \sec x \csc x\)

Tangent is Sine divided by Cosine

\(\ds \)

\(=\)

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

Secant is Reciprocal of Cosine

\(\ds \)

\(=\)

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

Cosecant is Reciprocal of Sine

\(\ds \)

\(=\)

\(\ds \frac 1 {\cos^2 x}\)

\(\ds \)

\(=\)

\(\ds \sec^2 x\)

Secant is Reciprocal of Cosine

$\blacksquare$