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Mathematics

Prove the following identity, where the angles involved are acute angles for which the expressions are defined.

(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A

Trigonometric Identities

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Answer

To prove:

(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A

Solving L.H.S. of the above equation :

⇒ (sin A + cosec A)2 + (cos A + sec A)2

⇒ sin2 A + cosec2 A + 2 sin A cosec A + cos2 A + sec2 A + 2 cos A sec A

⇒ sin2 A + cos2 A + cosec2 A + sec2 A + 2 sin A ×1sin A+ 2 cos A×1cos A\times \dfrac{1}{\text{sin A}} + \text{ 2 cos A} \times \dfrac{1}{\text{cos A}}

⇒ 1 + cosec2 A + sec2 A + 2 + 2

⇒ 5 + (1 + cot2 A) + (1 + tan2 A)

⇒ 7 + tan2 A + cot2 A.

Since, L.H.S. = R.H.S.

Hence, proved that (sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A.

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