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Trigonometric equations

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Question

If $\tan x + \cot x = 2$ and $x \in [0, 100 π]$ then number of values of x =

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Solution

$\tan x = 1 \Rightarrow x = n π + \frac{π}{4} = (4 n + 1) \frac{π}{4}$ , $n \in Z$
$\Rightarrow x = \frac{π}{4}, \frac{5 π}{4}, \frac{9 π}{4}, \frac{13 π}{4}, ..............$
$N o w 0 \leq (4 n + 1) \frac{π}{4} \leq 100 π$ $\Rightarrow 0 \leq 4 n + 1 \leq 400 \Rightarrow - 1 \leq 4 n \leq 399$
$\frac{- 1}{4} \leq n \leq 99.7$

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