Question

Prove that :

\[ \tan(-225^\circ)\cot(-405^\circ)-\tan(-765^\circ)\cot(675^\circ)=0 \]

Solution

We know that tangent and cotangent have period \(180^\circ\).

First,

\[ \tan(-225^\circ) = \tan(-225^\circ+180^\circ) \]

\[ = \tan(-45^\circ) = -1 \]

Next,

\[ \cot(-405^\circ) = \cot(-405^\circ+360^\circ) \]

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

Also,

\[ \tan(-765^\circ) = \tan(-765^\circ+720^\circ) \]

\[ = \tan(-45^\circ) = -1 \]

And,

\[ \cot(675^\circ) = \cot(675^\circ-540^\circ) \]

\[ = \cot135^\circ = -1 \]

Substituting these values,

\[ \begin{aligned} &\tan(-225^\circ)\cot(-405^\circ)-\tan(-765^\circ)\cot(675^\circ) \\[4pt] =& (-1)(-1)-(-1)(-1) \\[4pt] =& 1-1 \\[4pt] =& 0 \end{aligned} \]

Hence Proved.