4.2 If \( \sin 17^{\circ}=a \), WITHOUT using a calculator, express the following in ter of \( a \) : \( 4.2 .1 \tan 17^{\circ} \) \( 4.2 .2 \sin 107^{\circ} \) \( 4.2 .3 \cos ^{2} 253^{\circ}+\sin ^{2} 557^{\circ} \)

To express \( \tan 17^{\circ} \) in terms of \( a \), we can use the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Since \( \sin 17^{\circ} = a \), we need \( \cos 17^{\circ} \). From the Pythagorean identity, \( \cos^2 \theta = 1 - \sin^2 \theta \), we find \( \cos 17^{\circ} = \sqrt{1 - a^2} \). Therefore, \( \tan 17^{\circ} = \frac{a}{\sqrt{1 - a^2}} \). Next, for \( \sin 107^{\circ} \), we use the sine co-function identity. Since \( 107^{\circ} = 90^{\circ} + 17^{\circ} \), we have \( \sin 107^{\circ} = \cos 17^{\circ} \). Thus, \( \sin 107^{\circ} = \sqrt{1 - a^2} \). Finally, for \( \cos^2 253^{\circ} + \sin^2 557^{\circ} \), we note that \( 253^{\circ} = 270^{\circ} - 17^{\circ} \) and \( 557^{\circ} = 540^{\circ} + 17^{\circ} \). Hence, \( \cos 253^{\circ} = -\sin 17^{\circ} = -a \) and \( \sin 557^{\circ} = \sin 17^{\circ} = a \). Thus, we have \( \cos^2 253^{\circ} + \sin^2 557^{\circ} = (-a)^2 + a^2 = a^2 + a^2 = 2a^2 \).