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General Solution of Trigonometric Equation

If α ,β ,γ ar...

Question

If $α, β, γ$ are the angles made by a line with the positive directions of the coordinate axes, then $\frac{sin γ + cos β}{1 - sin α} =$

\frac{1 + sin α}{sin γ - 2 cos β}

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\frac{1 + {sin}^{2} α}{sin γ - cos β}

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\frac{1 + sin α}{sin γ - cos β}

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\frac{1 + 2 sin α}{sin γ - 2 cos β}

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Solution

The correct option is C $\frac{1 + sin α}{sin γ - cos β}$
We know that
${cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = 1$
$\Rightarrow {cos}^{2} α = {sin}^{2} γ - {cos}^{2} β$
$\Rightarrow 1 - {sin}^{2} α = (sin γ + cos β) (sin γ - cos β)$
$\Rightarrow (1 - sin α) (1 + sin α) = (sin γ + cos β) (sin γ - cos β)$
$\Rightarrow \frac{sin γ + cos β}{1 - sin α} = \frac{1 + sin α}{sin γ - cos β}$

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