$△ACB$ is a right triangle, right angled at $C$.

Therefore, by using Pythagoras theorem we get,

$A{B}^2=A{C}^2+B{C}^2$

$A{C}^2=A{B}^2-B{C}^2$

$A{C}^2={\left(29\right)}^2-{\left(21\right)}^2$

$A{C}^2=400$

$AC=20$ units

$\cos \theta =\frac{BC}{AB}=\frac{21}{29}$

$\sin \theta =\frac{AC}{AB}=\frac{20}{29}$

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