Question

$\bf \underline{Prove \: that:-}$
$\tt{cotx \: cot2x - cot2x \: cot3x \: - cot3x \: cotx \: = \: 1}$
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Answer 1

${\huge{\underbrace{\overbrace{\color{pink}{happy \: learning}}}}}$

Answer 2

Answer:

cot x cot 2x - cot 2x cot 3x - cot 3x cot 4x

= cot x cot 2x - cot 3x (cot 2x+ cot x)

= cot x cot 2x - cot (2x+x)(cot 2x+ cot x)

[ as cot(A+B)= cotAcotB cotAcotB−1 ]

= cot x cot 2x-( cotx+cot2xcot2xcotx−1 )(cot 2x+ cot x)

= cot x cot 2x- (cot 2x cot x-1)

= cot x cot 2x- cot 2x cot x+1

= 1

Hence, proved