SSC CHSL 2023 PYQ — Question 75

The correct answer is option 3 i.e. $\sqrt{16pq}=p^2-q^2$

cot A + cos A = p............1

cot A - cos A = q.............2

Add and subtract equations 1 and 2

2 cot A = p + q ..............3

2 cos A = p - q ...............4

multiply in equations 3 and 4

4 cotA.cosA = (p + q)(p - q)

⇒ 4 cotA.cosA = p² - q²       (square on both sides

⇒ cot²A.cos²A = (p² - q²)²/16...............5

square the equations 3 and 4 then subtract equation 4 from equation 3

⇒ (2 cot A)² - (2 cos A)²  = (p + q)² - (p - q)²

⇒ 4cos²A/sin²A - 4cos²A = p² + q² + 2pq - p² - q² + 2pq

⇒ 4cos²A{(1/sin²A) - 1}= 4pq

⇒ 4cos²A{(1 - sin²A)/sin²A}= 4pq

⇒ 4cos²A{cos²A/sin²A}= 4pq   

⇒ cos²A.cot²A = pq             {Put the value from equation 5}

⇒ (p² - q²)²/16 = pq 

⇒ (p² - q²)² = 16pq 

(p² - q²) = $\sqrt{16pq}$