If \(\sec^2\theta = 4\), then find the value of \(\sin^2\theta + \csc(90 - \theta)\) where \(0^\circ \lt \theta \lt 90^\circ\).
\(\dfrac{11}{4}\)
We are given \(\sec^2\theta = 4\) with \(0^\circ \lt \theta \lt 90^\circ\), so θ is acute.
Taking the positive square root (θ acute), \(\sec\theta = 2\), hence \(\cos\theta = \dfrac{1}{2}\).
This means \(\theta = 60^\circ\).
Now \(\sin\theta = \sin 60^\circ = \dfrac{\sqrt{3}}{2}\), so \(\sin^2\theta = \dfrac{3}{4}\).
Use the complementary identity \(\csc(90^\circ - \theta) = \sec\theta\).
So \(\csc(90^\circ - \theta) = \sec\theta = 2\).
Therefore \(\sin^2\theta + \csc(90^\circ - \theta) = \dfrac{3}{4} + 2 = \dfrac{3 + 8}{4} = \dfrac{11}{4}\).
Hence the value is \(\dfrac{11}{4}\).
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