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We have,
(i), (sin A + sec A)² + (cos A + cosec A)² = (1 + sec A.cosec A)
LHS = (sinA + secA)²+ ( cosA + cosecA)²
= sin²A + sec²A + 2sinAsecA + cos²A + cosec²A + 2cosA.cosecA
= sin²A + cos²A + sec²A + cosec²A + 2cotA + 2tanA
= 1 + (sin²A + cos²A)/sin²A.cos²A + 2 (cotA + tanA)
= 1 + 1/(sin²A.cos²A) + 2{(sin²A + cos²A)/(sinA + cosA)
= 1 + 2{1/(sinA.cosA)} + 1/(sin²A.cos²A)
= 1 + 2secA.cosecA + sec²A.cosec²A
= 1( + secA.cosecA)²
= RHS
Now the second one,
First let theta equat to @
(ii), ([email protected] + [email protected])²+ ([email protected] + [email protected])²= 7 + tan²@ + cot²@
LHS = ([email protected] + [email protected])²+ ([email protected] + [email protected])²
= sin²@ + cosec²@ + 2 + cos²@ + sec²@ + 2
= sin²@ + cos²@ + (1 + cot²@) + 1 + tan²@ + 4
= 1 + 6 + cot²@ + tan²@
= 7 + cot²@ + tan²@
= RHS
That's it
Hope it helped (+_+)