Question

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

Answer 1

Step-by-step explanation:

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Answer 2

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 : 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 : sec²A = 1 + tan²A

→ 1 + 1 + cot²A + 1 + tan²A = 4

→ 7 + cot²A + tan²A

= R.H.S.

Hence, proved !!