Prove that `2 + sec(x) cosec(x) = (sin x + cos x)^2 / (sin x cos x).`  

Expert Answers

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To prove, consider the left side of the equation.


Express the secant and cosecant in terms of cosine and sine, respectively.



To add, express them as two fractions with same denominators.


`=(2sinxcosx)/(sinxcosx) + 1/(sinxcosx)`

`=(2sinxcosx + 1)/(sinxcosx)`

Apply the Pythagorean identity `sin^2x+cos^2x=1` .



And, factor the numerator.

`= ((sinx +cosx)(sinx+cosx))/(sinxcosx)`


Notice that this is the same expression that the right side of the equation have. Thus, this proves that the  `2+secxcscx=(sinx+cosx)^2/(sinxcosx)`  is an identity.

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