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Prove that \(\frac{\sin \theta}{1-\cos \...

Prove that \(\frac{\sin \theta}{1-\cos \theta}\) = cosec θ + cot θ.

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The correct Answer is:
C
Given, Function `f(x)=x^(2), x in R`
` and g(A)={x in R: f(x) in A}: A subseteq R`
Now, for `S=[0,4]`
`g(S)={x in R: f(x) in S=[0,4]}`
`={x in R: x^(2) in [0,4]}`
`={x in R: x in [-2,2]}`
`rArr g(S)=[-2,2]`
So, `f(g(S))=[0,4]=S`
Now, `f(S)={x^(2): x in S =[0,14]} =[0,16]`
`and g(f(S))={x in R: f(x) in f(S)=[0,16]}`
`={x in R: f(x) in [0,16]}`
`={x in R: x^(2) in [0,16]}`
`={ x in R: x in [-4,4]}=[-4,4]`
From above, it is clear that `g(f(S))=g(S).`
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