Trigonometric Identities

The controlling idea is to rewrite every double-angle term as a squared single-angle expression so that a common factor emerges, using the three standard identities $1-\cos 2\theta =2{\sin }^2\theta$, $1+\cos 2\theta =2{\cos }^2\theta$, and $\sin 2\theta =2\sin \theta \cos \theta$. These conversions are what turn an opaque ratio of double-angle terms into something factorable. Substituting into the numerator gives $2{\sin }^2\theta +2\sin \theta \cos \theta$, which factors as $2\sin \theta (\sin \theta +\cos \theta )$. The denominator becomes $2{\cos }^2\theta +2\sin \theta \cos \theta$, factoring as $2\cos \theta (\cos \theta +\sin \theta )$. Both numerator and denominator now share the common factor $2(\sin \theta +\cos \theta )$, which cancels cleanly, leaving the simple ratio $\frac{\sin \theta }{\cos \theta }=\tan \theta$. Option $\cot \theta$ results from inverting the final ratio by swapping numerator and denominator. Option $\sin \theta$ drops the cosine in the denominator. Option $\sec \theta$ misidentifies which factor cancels. This matches the standard double-angle reduction widely used in JEE Advanced simplification problems. As a final plausibility check at $\theta =\frac{\pi }{4}$, where $\cos 2\theta =0$ and $\sin 2\theta =1$, the original expression evaluates to $\frac{1-0+1}{1+0+1}=1$, which equals $\tan \frac{\pi }{4}=1$, so the simplified form agrees with direct substitution and the result is confirmed.