Prove the identity \(\tan x + \frac{1}{\tan x} = \frac{1}{\sin x \cos x}\).
Solution
Start with the left-hand side (LHS):
\(\tan x + \frac{1}{\tan x} = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}\)
Combine the fractions:
\(\frac{\sin^2 x + \cos^2 x}{\sin x \cos x}\)
Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), the expression becomes:
\(\frac{1}{\sin x \cos x}\)
This matches the right-hand side (RHS), proving the identity.
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