Question : The value of $\tan ^2 48^{\circ}-\operatorname{cosec}^2 42^{\circ}+\operatorname{cosec}\left(67^{\circ}+\theta\right)-\sec \left(23^{\circ}-\theta\right)$ is:
Option 1: $-1$
Option 2: $0$
Option 3: $1$
Option 4: $-2$
Correct Answer: $-1$
Solution :
Given: $\tan ^2 48^{\circ}-\operatorname{cosec}^2 42^{\circ}+\operatorname{cosec}\left(67^{\circ}+\theta\right)-\sec \left(23^{\circ}-\theta\right)$
= $\tan ^2 48^{\circ}-\sec^2 (90^{\circ}-42^{\circ})+\sec(90^{\circ}-(67^{\circ}+\theta))-\sec \left(23^{\circ}-\theta\right)$
= $\tan ^2 48^{\circ}-\sec^2 48^{\circ}+\sec(23^{\circ}-\theta)-\sec \left(23^{\circ}-\theta\right)$
= $-1$
Hence, the correct answer is $-1$.
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