(cotA + secB)sqaure - (tanB -cosecA)square =2(cotA.secB + tanB.cosecA)
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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.
= 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.
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