Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of

$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$

$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$

In $\Delta ACB ,$ we have

$AC=\sqrt{ AB ^{2}- BC ^{2}}=\sqrt{(29)^{2}-(21)^{2}}$

$=\sqrt{(29-21)(29+21)}=\sqrt{(8)(50)}=\sqrt{400}=20$ units

So, $\sin \theta=\frac{A C}{A B}=\frac{20}{29}, \cos \theta=\frac{B C}{A B}=\frac{21}{29}$

Now,

$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta=\left(\frac{20}{29}\right)^{2}+\left(\frac{21}{29}\right)^{2}=\frac{20^{2}+21^{2}}{29^{2}}=\frac{400+441}{841}=1$

and

$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta=\left(\frac{21}{29}\right)^{2}-\left(\frac{20}{29}\right)^{2}=\frac{(21+20)(21-20)}{29^{2}}=\frac{41}{841}$