Prove that $c o t A c o t 2 A + c o t 2 A c o t 3 A + 2 = c o t A (c o t A - c o t 3 A)$

Open in App

Solution

cot A. cot 2A + 1 = $\frac{c o s (2 A - A)}{s i n A s i n 2 A} = \frac{c o s A}{s i n A s i n 2 A}$
Similarly, cot 2A cot 3A + 1 = $\frac{c o s A}{s i n 2 A s i n 3 A}$
$∴ L . H . S . = \frac{c o s A}{s i n 2 A} [\frac{1}{s i n A} + \frac{1}{s i n 3 A}]$
$∴ L . H . S . = \frac{c o s A}{s i n 2 A} [\frac{2 s i n 2 A c o s A}{s i n A s i n 3 A}]$
$= c o t A \cdot \frac{2 c o s A}{s i n 3 A} c o t A \cdot \frac{s i n 2 A}{s i n A s i n 3 A}$
$= c o t A \cdot \frac{s i n (3 A - A)}{s i n A s i n 3 A}$
$= c o t A (c o t A - c o t 3 A)$

Suggest Corrections

Similar questions

cot 3 A = \frac{3 cot A - {cot}^{3} A}{1 - 3 {cot}^{2} A}

$\frac{1}{t a n 3 A - t a n A}$ - $\frac{1}{c o t 3 A - c o t A}$ =

A + B + C = 180^{\circ}

(c o t B + c o t C) (c o t C + c o t A) (c o t A + c o t B)

\sqrt{2}

\frac{5 - 3 \sqrt{2}}{7}

\sqrt{2}

3 + 5 \sqrt{2}

Join BYJU'S Learning Program