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  • Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (sin A + cosec A)^ 2 + (cos A + sec A)^2 = 7 + tan^2 A + cot^2 A

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

    (viii)(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}= 7+\tan ^{2}A+\cot ^{2}A

           

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best_answer

Given equation,
(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}= 7+\tan ^{2}A+\cot ^{2}A..................(i)

Taking LHS;

(\sin A+\csc A)^{2}+(\cos A+\sec A)^{2}
\\\Rightarrow \sin^2 A+\csc^2A +2+\cos^2A+\sec^2A+2\\\\ \Rightarrow 1+2+2+(1+\cot^2A)+(1+\tan^2A)
[since \sin^2\theta +\cos^2\theta = 1, \csc^2\theta-\cot^2\theta =1, \sec^2\theta-\tan^2\theta=1]

\\7+\csc^2A+\tan^2A\\ =RHS

Hence proved
 

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manish

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