If A + B = pi/4 prove that (cot A-1) (cot B-1) = 2 - Maths - Trigonometric Functions

Question

If A + B = pi/4 prove that (cot A-1) (cot B-1) = 2

Lalit Mehra · Accepted Answer

Given, $A + B = \frac{π}{4}$

$∴ \cot (A+B) = \frac{π}{4} \Rightarrow \frac{\cot A cot B - 1}{\cot B + cot A} = \cot \frac{π}{4} \Rightarrow \frac{\cot A cot B - 1}{\cot B + cot A} = 1$

⇒ cot A cot B – 1 = cot B + cot A

⇒ cot A cot B – cot B – cot A = 1

⇒ cot A cot B – cot B – cot A + 1 = 1 + 1 (Adding 1 to both sides)

⇒ cot B (cot A – 1) – 1 (cot A – 1) = 2

⇒ (cot A – 1) (cot B – 1) = 2

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