First slide

Trigonometric equations

Question

The number of solutions of $\sec^{2} θ + {cosec}^{2} θ + 2 {cosec}^{2} θ = 8, 0 \leq θ \leq π / 2$ is

Moderate

A

4
B

3
C

0
D

2

Solution

We have
$\frac{1}{\sin^{2} {θcos}^{2} θ} + \frac{2}{\sin^{2} θ} = 8, \sin θ \neq 0, \cos θ \neq 0$
$\begin{array}{l} or 1 + 2 \cos^{2} θ = 8 \sin^{2} {θcos}^{2} θ \\ or 1 + 2 \cos^{2} θ = 8 \cos^{2} θ (1 - \cos^{2} θ) \\ or 8 \cos^{4} θ - 6 \cos^{2} θ + 1 = 0 \\ or (4 \cos^{2} θ - 1) (2 \cos^{2} θ - 1) = 0 \\ or \cos^{2} θ = 1 / 4 = \cos^{2} (π / 3) \\ or \cos^{2} θ = 1 / 2 = \cos^{2} (π / 4) \\ or θ = nπ \pm (π / 3) or θ = nπ \pm (π / 4), n \in Z \end{array}$
Hence, for $0 \leq θ \leq π / 2, θ = π / 3, θ = π / 4$

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