Question

Given a ∆ABC, in which ∠C = 90°, ∠ABC = θ°, BC = 21 units, AB = 29 units.
Show that (cos²θ − sin²θ) = $\frac{41}{841}.$

Answer 1

Using Pythagoras theorem, we get:
AB² = AC² + BC²
⇒ AC² = AB² $-$ BC²
⇒ AC² = (29)² $-$ (21)²
⇒ AC² = 841$-$ 441
⇒ AC² = 400
⇒ AC = $\sqrt{400}$ = 20 units

Now, sin $\theta =\frac{AC}{AB}=\frac{20}{29}$ and cos $\theta$ = $\frac{BC}{AB}=\frac{21}{29}$

cos² $\theta$ $-$ sin² $\theta$ = ${\left(\frac{21}{29}\right)}^2-{\left(\frac{20}{29}\right)}^2=\frac{441}{841}-\frac{400}{841}=\frac{41}{841}$

Hence Proved.