Prove the following- sin- -cos- tan- -cot- -sec- -cosec-

To prove (sinα +cosα )(tanα +cotα )=secα +cosecα.Consider L.H.S:LHS=(sinα +cosα )(tanα +cotα )0ex0ex =(sinα +cosα )(sinα/cosα+cosα/sinα) [∵tanα =si ...

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Standard XII

Mathematics

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

Prove the fol...

Question

Prove the following: $(\sin α + \cos α) (\tan α + \cot α) = \sec α + cosec α$

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Solution

To prove $(\sin α + \cos α) (\tan α + \cot α) = \sec α + cosec α$ .
Consider L.H.S:
$LHS = (\sin α + \cos α) (\tan α + \cot α) = (\sin α + \cos α) (\frac{\sin α}{\cos α} + \frac{\cos α}{\sin α}) [∵ \tan α = \frac{\sin α}{\cos α} and \cot α = \frac{\cos α}{\sin α}] = (\sin α + \cos α) (\frac{\sin^{2} α + \cos^{2} α}{\sin α \cdot \cos α}) = (\sin α + \cos α) (\frac{1}{\sin α \cdot \cos α}) [∵ \sin^{2} α + \cos^{2} α = 1] = \frac{\sin α + \cos α}{\sin α \cdot \cos α} = \frac{\sin α}{\sin α \cdot \cos α} + \frac{\cos α}{\sin α \cdot \cos α} = \frac{1}{\cos α} + \frac{1}{\sin α} = \sec α + \cos e c α [∵ \frac{1}{\cos α} = s e c α and \frac{1}{\sin α} = \cos e c α] = RHS$
Thus $LHS = RHS$
Hence, $(\sin α + \cos α) (\tan α + \cot α) = \sec α + cosec α$ is proved.

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Prove the following: (sinα+cosα)(tanα+cotα)=secα+cosecα

Prove the following: $(\sin α + \cos α) (\tan α + \cot α) = \sec α + cosec α$