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Question
Prove the following identities:
sin A (1 + tan A) + cos A (1 + cot A) = sec A + coseс А
Theorem
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Solution
LHS = `sin A(1 + (sin A)/(cos A)) + cos A(1 + (cos A)/(sin A))`
= `(sin A (cos A + sin A))/(cos A) + (cos A (sin A + cos A))/(sin A)`
= `(sin A + cos A)((sin A)/(cos A) + (cos A)/(sin A))`
= `((sin A + cos A)(sin^2A + cos^2A))/(sin A cos A)`
= `((sin A + cos A))/(sin A cos A)`
= `((sin A)/(sin A cos A) + (cos A)/(sin A cos A))`
= `(1/(cos A) + 1/(sin A))`
= sec A + cosec A = RHS
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