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General Solution of Trigonometric Equation

If α +β = 5...

Question

If $α + β = \frac{5 π}{4}$ , then value of $\frac{cot α \cdot cot β}{(1 + cot α) (1 + cot β)}$ is (wherever defined)-

A

1

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B

\frac{3}{2}

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C

\frac{1}{2}

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D

\frac{5}{4}

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Solution

The correct option is D $\frac{1}{2}$
Given $α + β = \frac{5 π}{4}$
$⟹ cot (α + β) = cot \frac{5 π}{4}$
$⟹ \frac{cot α cot β - 1}{cot α + cot β} = 1$
$⟹ cot α cot β - 1 = cot α + cot β$
$⟹ cot α cot β = 1 + cot α + cot β$
$⟹ 2 cot α cot β = 1 + cot α + cot β + cot α cot β$
$⟹ 2 cot α cot β = (1 + cot α) (1 + cot β)$
$⟹ \frac{cot α cot β}{(1 + cot α) (1 + cot β)} = \frac{1}{2}$

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α + β = \frac{5 π}{4}

, then value of

\frac{cot α cot β}{(1 + cot α) (1 + cot β)}

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α + β = \frac{5 π}{4} .

Then the value of

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α + β = \frac{5 π}{4},

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α + β = \frac{5 π}{4}

, then the value of

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