Question

Prove or disprove that 2cosx is the same as cotx(cosxtanx+sinx)

Answer 1

We'll use the following definitions of trigonometric identities to prove the statement:

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

With this, we start rewriting the left given expression:

$cot(x)[cos(x)tan(x)+sin(x)]=$
$\frac{cos(x)}{sin(x)}[cos(x) \frac{sin(x)}{cos(x)} +sin(x)]=$
$\frac{cos(x)}{sin(x)}[sin(x) +sin(x)]= \frac{cos(x)}{sin(x)}[2sin(x)]=2cos(x)$

Hence, we have proven that the statement is true.