Let $PQR$ be a right-angled triangle, right-angled at $R$. Given $PQ = 29 ext{ cm}$, $QR = 21 ext{ cm}$, and $\angle Q = heta$. Find the value of $\cos^2 heta - \sin^2 heta$.

Question

Let $PQR$ be a right-angled triangle, right-angled at $R$. Given $PQ = 29 	ext{ cm}$, $QR = 21 	ext{ cm}$, and $\angle Q = 	heta$. Find the value of $\cos^2 	heta - \sin^2 	heta$.

Accepted Answer

**Solution:**

1. Given triangle $PQR$ is right-angled at $R$, with sides:
   - $PQ = 29 	ext{ cm}$ (hypotenuse)
   - $QR = 21 	ext{ cm}$
   - $PR = ?$

2. Using Pythagoras theorem:
   $$PR^2 = PQ^2 - QR^2 = 29^2 - 21^2 = 841 - 441 = 400$$
   $$\Rightarrow PR = \sqrt{400} = 20 	ext{ cm}$$

3. Calculate $\cos 	heta$ and $\sin 	heta$ where $	heta = \angle Q$:
   - $\cos 	heta = \frac{QR}{PQ} = \frac{21}{29}$
   - $\sin 	heta = \frac{PR}{PQ} = \frac{20}{29}$

4. Use the identity:
   $$\cos^2 	heta - \sin^2 	heta = (\cos 	heta)^2 - (\sin 	heta)^2$$

5. Substitute values:
   $$= \left(\frac{21}{29}ight)^2 - \left(\frac{20}{29}ight)^2 = \frac{441}{841} - \frac{400}{841} = \frac{441 - 400}{841} = \frac{41}{841}$$

**Answer:** $\frac{41}{841}$

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