Question

(v)cos 7a+cos3a-cos 5a- cosa/sin 7a -sin 3a -sin 5a + sina= cot 2a​

Answer 1

Answer:

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Answer 2

$\huge \bold \color{green}{ \underline{ \underline \red{required \: answer : }}}$

$\bold{ \underline{ \underline{LHS }}} = \frac{ \cos7A + \cos3A - \cos5A - \cos \: A}{ \sin7A - \sin3A \: - \sin5A + \sin \: A}$

$\bold{ \underline{ \underline{RHS}}} = \cot2A$

$\huge\bold{{ \underline { \underline{solution}}}}$

first we consider LHS

$\bold{ = \frac{ \cos7A + \cos3A - \cos5A - \cos \: A}{ \sin7A - \sin3A \: - \sin5A + \sin \: A}}$

$\bold{= \frac{ - 2 \sin4A\sin3A + 2 \sin4A \sin \: A}{2 \sin4A \cos3A - 2 \sin4A \cos \: A}}$

$= \frac{ - \sin4A( \sin3A - \sin \: A)}{ \sin4A( \cos3A - \cos \: A)}$

$= \frac{ - \sin3A - \sin \: A}{ \cos3A - \cos \: A}$

$= \frac{ - 2 \cos2A \sin \: A}{2 \sin2A \sin\:( - A)}$

$= \frac{ - 2 \cos2A \sin \: A}{ - 2 \sin2A \sin \: A}$

$= \frac{ \cos2A}{ \sin2A}$

$= \cot2A$

$\huge \bold{so \: LHS = RHS}$

$\bold \color{blue}{hence \: proved}$