(sinA +cosecA )2 +(cosA+secA ) 2 =7+tan2A+cot2A
Answers
Step-by-step explanation:
To Prove : (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A
L.H.S. = (sin A + cosec A)² + (cos A + sec A)²
- Identity : (a + b)² = a² + b² + 2ab
→ [ (sin A)² + (cosec A)² + 2(sin A)(cosec A) ] + [ (cos A)² + (sec A)² + 2(cos A)(sec A) ]
→ [ sin²A + cosec²A + 2(sin A)(cosec A) ] + [ cos²A + sec²A + 2(cos A)(sec A) ]
- Identity : cosec A = 1/sin A
- Identity : cosec A = 1/sin A Identity : sec a = 1/cos A
→ [ sin²A + cosec²A + 2(sin A)(1/sin A) ] + [ cos²A + sec²A + 2(cos A)(1/cos A) ]
→ [ sin²A + cosec²A + 2 ] + [ cos²A + sec²A + 2 ]
- Opening the brackets.
→ sin²A + cosec²A + 2 + cos²A + sec²A + 2
- Rearranging the terms.
→ sin²A + cos²A + cosec²A + sec²A + 2 + 2
→ sin²A + cos²A + cosec²A + sec²A + 4
- Identity : sin²A + cos²A = 1
→ 1 + cosec²A + sec²A + 4
- Identity : cosec²A = 1 + cot²A
- Identity : cosec²A = 1 + cot²AIdentity : sec²A = 1 + tan²A
→ 1 + 1 + cot²A + 1 + tan²A = 4
→ 7 + cot²A + tan²A
= R.H.S.
Hence, proved !!
Other identities which are frequently used :
• tan A = sin A/cos A
• cot A = cos A/sin A
• cot A = 1/tan A
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