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Trigonometric Ratios

Prove that: x...

Question

Prove that: $cot x cot 2 x - cot 2 x cot 3 x - cot 3 x cot x = 1$

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Solution

$cot x cot 2 x - cot 2 x cot 3 x - cot 3 x cot x = 1$
$L . H . S = cot x cot 2 x - cot 2 x cot 3 x - cot 3 x cot x$
$= cot x cot 2 x - cot 3 x (cot 2 x + cot x)$
$= cot x cot 2 x - cot (2 x + x) (cot 2 x + cot x)$
$cot (A + B) = \frac{cot A cot B - 1}{cot A + cot B}$
$= cot x cot 2 x - (\frac{cot 2 x cot x - 1}{cot 2 x + cot x}) (cot 2 x + cot x)$
$= cot x cot 2 x - (cot 2 x cot x - 1) = 1$
$\Rightarrow L . H . S = R . H . S .$
Hence proved

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Trigonometric Ratios

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