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Sin(A+B)Sin(A-B)

Prove that co...

Question

Prove that $(\frac{\cos A}{1 + \sin A}) + (\frac{1 + \sin A}{\cos A}) = 2 s e c A$

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Solution

Consider, $L . H . S = (\frac{\cos A}{1 + \sin A}) + (\frac{1 + \sin A}{\cos A})$
$= \frac{\{\cos^{2} A + {(1 + \sin A)}^{2}\}}{\cos A . (1 + \sin A)}$
$= \frac{\cos^{2} A + [{(1)}^{2} + 2 (1) (\sin A) + {(\sin A)}^{2}]}{\cos A . (1 + \sin A)}$
$= \frac{\cos^{2} A + [1 + 2 \sin A + \sin^{2} A]}{\cos A . (1 + \sin A)}$
$= \frac{\cos^{2} A + \sin^{2} A + 1 + 2 \sin A}{\cos A . (1 + \sin A)}$
$= \frac{1 + 1 + 2 \sin A}{\cos A . (1 + \sin A)}$ [ $\sin ² A + \cos ² A = 1$ ]
$= \frac{2 + 2 \sin A}{\cos A . (1 + \sin A)}$
$= \frac{2 (1 + \sin A)}{\cos A . (1 + \sin A)}$
$= \frac{2}{\cos A}$
$= 2 s e c A$ [ $\frac{1}{\cos A} = s e c A$ ]
$= R . H . S .$
Hence it is proved that, $(\frac{\cos A}{1 + \sin A}) + (\frac{1 + \sin A}{\cos A}) = 2 s e c A$

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Thousands	Hundreds	Tens	Ones	Tenths	Hundredths	Thousandths
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\frac{c o s A}{1 + s i n A} + \frac{1 + s i n A}{c o s A} = 2 s e c A

Prove the identity cos2A/1+sinA+1+sinA/cosA=2secA

Prove that:

$\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 S e c A$

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Sin(A+B)Sin(A-B)

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