√[(1 - cos A) / (1 + cos A)] = sin A / (1 + cos A)

The L.H.S. of the equation can be written as, Since, L.H.S. = R.H.S. hence, proved that

Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
$\sqrt{\dfrac{1 - \text{cos A}}{\text{1 + cos A}}} = \dfrac{\text{sin A}}{\text{1 + cos A}}$

Trigonometric Identities

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Answer

The L.H.S. of the equation can be written as,

$\Rightarrow \sqrt{\dfrac{(1 - \text{cos A})(1 + \text{cos A})}{\text{(1 + cos A)(1 + \text{cos A})}}} \\[1em] \Rightarrow \sqrt{\dfrac{(1 - \text{cos}^2 A)}{\text{(1 + cos A)}^2}} \\[1em] \Rightarrow \sqrt{\dfrac{\text{sin}^2 A}{\text{(1 + cos A)}^2}} \\[1em] \Rightarrow \dfrac{\text{sin A}}{\text{1 + cos A}}.$

Since, L.H.S. = R.H.S. hence, proved that $\sqrt{\dfrac{1 - \text{cos A}}{\text{1 + cos A}}} = \dfrac{\text{sin A}}{\text{1 + cos A}}$

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Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
$\sqrt{\dfrac{1 - \text{cos A}}{\text{1 + cos A}}} = \dfrac{\text{sin A}}{\text{1 + cos A}}$

Trigonometric Identities

Answer

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Mathematics

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Trigonometric Identities

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Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
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Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:
$\dfrac{\text{tan}^3 \text{ θ} - 1}{\text{tan θ} - 1} = \text{sec}^2 \text{ θ} + \text{tan θ}.$

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$\sqrt{\dfrac{\text{1 - cos A}}{\text{1 + cos A}}} = \text{cosec A - cot A}$