First slide

Trigonometric equations

Question

The equation $\tan^{2} x + \cot^{2} x = 4 a {cosec}^{2} 2 x$ has $a$ real solution if

Moderate

A

$0 < a \leq 1$
B

$1 / 2 \leq a \leq 1$
C

$1 / 4 \leq a \leq 1 / 2$
D

$- 1 \leq a \leq 1$

Solution

$\begin{array}{l} a = {(\sin^{2} x + \cos^{2} x)}^{2} - 2 \sin^{2} x \cos^{2} x \\ \Rightarrow \sin^{2} 2 x = - 2 (a - 1) \Rightarrow 1 - \cos 4 x = - 4 (a - 1) \\ \Rightarrow \cos 4 x - 4 a - 3 so - 1 \leq 4 a - 3 \leq 1 \\ \Rightarrow 1 / 2 \leq a \leq 1 \end{array}$

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