Question

Prove that :

\[ \tan225^\circ \cot405^\circ + \tan765^\circ \cot675^\circ = 0 \]

Solution

\[ \tan225^\circ = \tan(180^\circ+45^\circ) = \tan45^\circ = 1 \]

\[ \cot405^\circ = \cot(360^\circ+45^\circ) = \cot45^\circ = 1 \]

\[ \tan765^\circ = \tan(720^\circ+45^\circ) = \tan45^\circ = 1 \]

\[ \cot675^\circ = \cot(720^\circ-45^\circ) = \cot315^\circ = -1 \]

Therefore,

\[ =(1)(1)+(1)(-1) \]

\[ =1-1 \]

\[ =0 \]

Hence Proved.