Question:
cosec257° – tan233° = ?
(a) 1
(b) 0
(c) –1
(d) 2
Solution:
$\operatorname{cosec}^{2} 57^{\circ}-\tan ^{2} 33^{\circ}$
$=\left(\operatorname{cosec}\left(90^{\circ}-33^{\circ}\right)\right)^{2}-\tan ^{2} 33^{\circ}$
$=\sec ^{2} 33^{\circ}-\tan ^{2} 33^{\circ} \quad\left(\because \operatorname{cosec}\left(90^{\circ}-\theta\right)=\sec \theta\right)$
$=1$ (using the identity : $\sec ^{2} \theta-\tan ^{2} \theta=1$ )
Hence, the correct option is (a).