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

sin2θ+sin2θcos2θ>0⇒ sin2θ(cos2θ+1)cos2θ>0⇒ Tan2θ(1+cos2θ)>0⇒ 2θ∈0,π2∪π2,3π2∪2π,5π2∪3π,7π2⇒ θ∈0,π4∪π2,3π4∪π,5π4∪3π2,7π4

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

sin2θ+sin2θcos2θ>0⇒ sin2θ(cos2θ+1)cos2θ>0⇒ Tan2θ(1+cos2θ)>0⇒ 2θ∈0,π2∪π2,3π2∪2π,5π2∪3π,7π2⇒ θ∈0,π4∪π2,3π4∪π,5π4∪3π2,7π4

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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

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