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2015-10-18T19:36:57+05:30
LHS = (cotA + secB)² - (tanB -cosecA)²
= cot²A + sec²B + 2cotAsecB - (tan²B + cosec²A - 2tanBcosecA
= cot²A + sec²B + 2cotAsecB - tan²B - cosec²A + 2tanBcosecA
= cot²A - cosec²A + sec²B - tan²B + 2cotAsecB + 2tanBcosecA
= -(cosec²A - cot²A) + (sec²B - tan²B) + 2(cotAsecB + tanBcosecA)
= -1 + 1 + 2(cotAsecB + tanBcosecA)
= 2(cotAsecB + tanBcosecA)
= RHS

Proved.
2 5 2