Prove that √sec^2 A + cosec^2 A = tanA + cotA
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√sec²A + cosec²A = tanA + cotA
LHS:
√sec²A + cosec²A = √1 + tan²A + 1 + cot²A
∵ sec²Ф = 1 + tan²Ф
cosec ²Ф = 1 + cot²Ф
∴√sec²A + cosec²A = √tan²A + 2 + cot²A
= √tan²A + 2( tanA × cotA ) + cot²A
= √ ( tanA + cotA )²
∵ tanФ × cotФ = 1
√sec²A + cosec²A = tanA + cotA
= RHS
LHS:
√sec²A + cosec²A = √1 + tan²A + 1 + cot²A
∵ sec²Ф = 1 + tan²Ф
cosec ²Ф = 1 + cot²Ф
∴√sec²A + cosec²A = √tan²A + 2 + cot²A
= √tan²A + 2( tanA × cotA ) + cot²A
= √ ( tanA + cotA )²
∵ tanФ × cotФ = 1
√sec²A + cosec²A = tanA + cotA
= RHS
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