If cot ⁡A + cos ⁡A = p, cot ⁡A - cos ⁡A = q, then what is the value of p²- q²?​​

Correct option is A

Given: ​

cot ⁡A + cos ⁡A = p

cot ⁡A - cos ⁡A = q

Formula Used: ​

a² - b² = (a + b)(a - b) 

Solution: ​

p²- q² = (p + q)(p - q) 

= (cot ⁡A + cos ⁡A + cot ⁡A - cos ⁡A)(cot ⁡A + cos ⁡A - cot ⁡A + cos ⁡A) 

= (2cot A) (2cos A) 

= 4 (cot A cos A)

​$$\sqrt{pq} = \sqrt{(\cot A + \cos A)(\cot A - \cos A)} \\ \ \\ = \sqrt{\cot^2 A - \cos^2 A} \\ \ \\ = \sqrt{\frac{\cos^2A}{\sin^2A} - \cos^2 A} \\ \ \\ = \sqrt{\frac{\cos^2A -\sin^2A\cdot \cos^2 A }{\sin^2A}} \\ \ \\ = \sqrt{\frac{\cos^2A(1 -\sin^2A)} {\sin^2A}} \\ \ \\ = \sqrt{\cot^2A(\cos^2A)} \\ \ \\ = \cot A(\cos A)$$

if we multiply by 4, we get 

​$$4\sqrt {pq} = 4(\cot A \cdot \cos A)$$ 

​p²- q² = $$4\sqrt{pq}$$

Alternate Solution(short trick):