Question

(cosecA-sinA)(secA-cosA)(tanA+cotA)=1


prove that

Answer 1

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Answer 2

Answer:  The proof of the given equality is done below.

Step-by-step explanation:  We are given to prove the following trigonometric equality :

$(\csc A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1.$

We will be using the following trigonometric identities :

$(i)~\csc A=\dfrac{1}{\sin A},\\\\\\(ii)~\sec A=\dfrac{1}{\cos A},\\\\\\(iii)~\cos^2 A+\sin^2 A=1,\\\\\\(iv)\tan A=\dfrac{\sin A}{\cos A},\\\\\\(v)~\cot A=\dfrac{\cos A}{\sin A}.$

The proof is as follows :

$L.H.S.\\\\=(\csc A-\sin A)(\sec A-\cos A)(\tan A+\cot A)\\\\\\=\left(\dfrac{1}{\sin A}-\sin A\right)\left(\dfrac{1}{\cos A}-\cos A\right)\left(\dfrac{\sin A}{\cos A}+\dfrac{\cos A}{\sin A}\right)\\\\\\=\dfrac{1-\sin^2A}{\sin A}.\dfrac{1-\cos^2A}{\cos A}.\dfrac{\sin^2A+\cos^2A}{\sin A\cos A}\\\\\\=\dfrac{\cos^2A}{\sin A}.\dfrac{\sin^2A}{\cos A}.\dfrac{1}{\sin A\cos A}\\\\\\=1\\\\=R.H.S.$

Thus, we get

$(\csc A-\sin A)(\sec A-\cos A)(\tan A+\cot A)=1.$

Hence proved.