Question

Prove $\tan (-{225}^o)\cot (-{405}^o)-\tan (-{765}^0)\cot \left({675}^o\right)=0$

Answer 1

$\tan (-225{)}^{\circ }\cot (-405{)}^{\circ }-\tan (-765{)}^{\circ }\cot (675{)}^{\circ }$

$=-\tan {225}^{\circ }(-\cot {405}^{\circ })+\tan {765}^{\circ }\cot {675}^{\circ }$

$=\tan (180+45{)}^{\circ }\cot (360+45{)}^{\circ }+\tan (360\times 2+45{)}^{\circ }\cot (360\times 2-45{)}^{\circ }$

$=\tan {45}^{\circ }\cot {45}^{\circ }+\tan {45}^{\circ }(-\cot {45}^{\circ })$

$=\tan {45}^{\circ }\times \frac{1}{\tan {45}^{\circ }}-\tan {45}^{\circ }\times \frac{1}{\tan {45}^{\circ }}$

$=0$