Question

Prove that (Sin A + CosecA)2 + (Cos A + Sec A)2 = 7 + Tan A + Cot A.

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Answer 1

Answer:

Question:- Prove that: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A

Identities Used:-

1 + tan²A = sec²A  

1 + cot²A = cosec²A  

sin²A + cos²A = 1

cosecA = $\frac{1}{sinA}$

secA = $\frac{1}{cosA}$

Answer:-

= (sinA + cosecA)²+ (cosA + secA)²  

= sin²A + cosec²A + 2sinA cosecA + cos²A + sec²A + 2cosA secA  

= sin²A + cos²A + cosec²A + sec²A + 2sinA × $\frac{1}{sinA}$ + 2cosA× $\frac{1}{cosA}$  

= 1 + cosec²A + sec²A + 2 + 2  

= 5 + (1+cot²A) + (1+tan²A)  

= 7 + tan²A + cot²A

Hence Proved.