PYQ - Prove the given equation — task. Maths CBSE Live product, Class 10.

Prove that:

\frac{\sin θ}{1 + \cos θ} + \frac{1 + \cos θ}{\sin θ} = 2 cosec θ

Proof:

L.H.S \(=\)

\frac{\sin θ}{1 + \cos θ} + \frac{1 + \cos θ}{\sin θ}

\(LHS=\)

By applying \((a+b)^2\) identity, then we get,

\(LHS=\)

By applying the known identity then we get,

By simplyfing this then we get,

\(= 2 cosec \theta\)

\(=\) R.H.S

Hence proved.

Answer variants:

= \frac{\sin^{2} θ + {(1 + \cos θ)}^{2}}{\sin θ (1 + \cos θ)}

= \frac{2 + 2 \cos θ}{\sin θ (1 + \cos θ)}

= \frac{\sin^{2} θ + \cos^{2} θ + 1 - 2 \cos θ}{\sin θ (1 + \cos θ)}

= \frac{\sin^{2} θ + \cos^{2} θ + 1 + 2 \cos θ}{\sin θ (1 + \cos θ)}

= \frac{2 + \cos θ}{\sin θ (1 + \cos θ)}

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