Question

33. Prove that
(sin A + cosec A)? + (cos A + sec A)? = 7 + tan? A + cot? A​

Answer 1

Correct Question

To prove

(sinA + cosecA)² + (cosA + secA)² = 7 + tan²A + cot²A

Solution

Taking LHS,

(sinA + cosecA)² + (cosA + secA)²

→ sin²A + cosec²A + 2sinAcosecA + cos²A + sec²A + 2cosAsecA

(using, (a + b)² = a² + b² + 2ab)

→ (sin²A + cos²A) + cosec²A + sec²A + 2 + 2

(using, sinAcosecA = 1 and cosAsecA = 1)

→ 1 + cosec²A + sec²A + 4

(using, cos²A + sin²A = 1)

Now, we know that, cosec²A = 1 + cot²A and sec²A = 1 + tan²A

so,

→ 1 + (1 + cot²A) + (1 + tan²A) + 4

→ 7 + tan²A + cot²A

= RHS

Hence proved.