Prove the following: Sin (50 degree …

LHS= {tex}\sin \left( 50 ^ { \circ } + \theta \right) - \cos \left( 40 ^ { \circ } - \theta \right) + \tan 1 ^ { \circ } \tan 10 ^ { \circ } \tan 20 ^ { \circ } \tan 70 ^ { \circ }{/tex}{tex}\tan 80 ^ { \circ } \tan 89 ^ { \circ }{/tex}
{tex}= \sin \left[ 90 ^ { \circ } - \left( 40 ^ { \circ } - \theta \right) ] - \cos \left( 40 ^ { \circ } - \theta \right) + \tan 1 ^ { \circ } \tan 10 ^ { \circ } \tan 20 ^ { \circ }\right. \tan \left( 90 ^ { \circ } - 20 ^ { \circ } \right) \tan \left( 90 ^ { \circ } - 10 ^ { \circ } \right) \tan \left( 90 ^ { \circ } - 1 ^ { \circ } \right){/tex}
{tex}= \cos \left( 40 ^ { \circ } - \theta \right) - \cos \left( 40 ^ { \circ } - \theta \right) + \tan 1 ^ { \circ } \tan 10 ^ { \circ }\tan 20 ^ { \circ } \cot 20 ^ { \circ } \cot 10 ^ { \circ } \cot 1 ^ { \circ }{/tex}
{tex}= 0 + 1\\ = 1\\= RHS{/tex}