Question

(cosec A -Sin A)(Sec A - COS A)(tan A +cot A)= 1

Answer 1

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Answer 2

To prove :-

(cosec A -Sin A)(Sec A - COS A)(tan A +cot A)= 1

Proof :-

Here, L.H.S. = (cosec A -Sin A)(Sec A - COS A)(tan A +cot A)

And R.H.S. = 1

Solving L.H.S.,

=> (cosec A -Sin A)(Sec A - COS A)(tan A +cot A)

$=> (\frac{1}{sin\:\:A} - sin \:\:A)(\frac{1}{cos\:\:A} - cos\:\:A)(\frac{sin\:\:A}{cos\:\:A}+\frac{cos\:\:A}{sin\:\:A} )$

$\\ \\ (cosec=\frac{1}{sin}; \:\:sec = \frac{1}{cos}; \:\:tan=\frac{sin}{cos};\:\: cot= \frac{cos}{sin})\\ \\ \\=> (\frac{1-sin^{2}A }{sin\:\:A} )(\frac{1-cos^{2}A }{cos\:\:A})(\frac{1-sin^{2}A+cos^{2} A}{sin\:\:A \times cos\:\:A}) \\ \\ \\=> \frac{cos^{2} A}{sin\:\:A} \times \frac{sin^{2} A}{cos\:\:A} \times \frac{1}{sin\:\:A \times cos\:\:A}$

$[sin^{2}+cos^{2} =1] \\ \\ \\ => \frac{cos^{2}A \times sin^{2}A}{(sin\:\:A \times cos\:\:A \times sin\:\:A \times cos\:\:A)} \\ \\ \\=> \frac{cos^{2}A \times sin^{2}A}{sin^{2} A \times cos^{2} A} \\ \\ \\=> 1 = R.H.S.$

Hence, proved.