Question

Verify the identity:

1/Tanx +Tanx = Sec^2x/Tanx

Answer 1

\[
\frac{1}{tan(x)} + tan(x) = \large \frac{\quad1\quad}{ \frac{sin(x)}{cos(x)}} \normalsize + \frac{sin(x)}{cos(x)} = \\
= \frac{cos(x)}{sin(x)} + \frac{sin(x)}{cos(x)} = \frac{cos^2(x) + sin^2(x)}{sin(x)cos(x)} = \frac{1}{sin(x)cos(x)} \\
\frac{sec^2(x)}{tan(x)} = \large \frac{\;\; \frac{1}{cos^2(x)}\;\;}{ \frac{sin(x)}{cos(x)} } \normalsize = \frac{cos(x)}{sin(x)cos^2(x)} = \frac{1}{sin(x)cos(x)} \\
\frac{1}{tan(x)} + tan(x) = \frac{sec^2(x)}{tan(x)}
\]