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Question
Prove that: tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
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Solution
LHS =\[ \tan \left( - 225^\circ \right) \cot \left( - 405^\circ \right) - \tan \left( - 765^\circ \right) \cot \left( 675^\circ \right)\]
\[ = \left[ - \tan \left( 225^\circ \right) \right]\left[ - \cot \left( 405^\circ \right) \right] - \left[ - \tan \left( 765^\circ \right) \right] \cot \left( 675^\circ \right) \left[ \because \tan \left( - x \right) = \tan \left( x \right) and \cot \left( - x \right) = - \cot \left( x \right) \right]\]
\[ = \tan \left( 225^\circ \right) \cot \left( 405^\circ \right) + \tan \left( 765^\circ \right) \cot \left( 675^\circ \right)\]
\[ = \tan \left( 90^\circ \times 2 + 45^\circ \right) \cot \left( 90^\circ \times 4 + 45^\circ \right) + \tan \left( 90^\circ \times 8 + 45^\circ \right) \cot \left( 90^\circ \times 7 + 45^\circ \right)\]
\[ = \tan \left( 45^\circ \right) \cot \left( 45^\circ \right) + \tan \left( 45^\circ \right)\left[ - \tan \left( 45^\circ \right) \right]\]
\[ = 1 \times 1 + 1 \times \left( - 1 \right)\]
\[ = 1 - 1\]
\[ = 0\]
RHS
Hence, proved .
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