All possible values of θ∈0,2πfor which sin2θ+tan2θ>0 lie in:

0 , π 4 ∪ π 2 , 3 π 4 ∪ π , 5 π 4 ∪ 3 π 2 , 7 π 4

All possible values of θ∈0,2πfor which sin2θ+tan2θ>0 lie in:

0 , π 4 ∪ π 2 , 3 π 4 ∪ π , 5 π 4 ∪ 3 π 2 , 7 π 4

0 , π 4 ∪ π 2 , 3 π 4 ∪ 3 π 2 , 11 π 6

0 , π 2 ∪ π 2 , 3 π 4 ∪ π , 7 π 6

0 , π 4 ∪ π 2 , 3 π 4 ∪ π , 5 π 4 ∪ 3 π 2 , 7 π 4

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All possible values of $θ \in [0, 2 π]$ for which $\sin 2 θ + \tan 2 θ > 0$ lie in:

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$(0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (\frac{3 π}{2}, \frac{11 π}{6})$

$(0, \frac{π}{2}) \cup (π, \frac{3 π}{2})$

$(0, \frac{π}{2}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (π, \frac{7 π}{6})$

$(0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (π, \frac{5 π}{4}) \cup (\frac{3 π}{2}, \frac{7 π}{4})$

answer is D.

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Detailed Solution

$\begin{array}{l} \sin 2 θ + \frac{\sin 2 θ}{\cos 2 θ} > 0 \Rightarrow \frac{\sin 2 θ (\cos 2 θ + 1)}{\cos 2 θ} > 0 \\ \Rightarrow Tan 2 θ (1 + \cos 2 θ) > 0 \\ \Rightarrow 2 θ \in (0, \frac{π}{2}) \cup (\frac{π}{2}, \frac{3 π}{2}) \cup (2 π, \frac{5 π}{2}) \cup (3 π, \frac{7 π}{2}) \\ \Rightarrow θ \in (0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4}) \cup (π, \frac{5 π}{4}) \cup (\frac{3 π}{2}, \frac{7 π}{4}) \end{array}$