First slide

Compound angles

Question

Let $α, β, and γ$ be some angles in the first quadrant satisfying $\tan (α + β) = 15 / 8$ and $cosec γ = 17 / 8$ , then which of the following hold(s) good?

Moderate

A

$α + β + γ = π$
B

$\cot αcot βcot γ = \cot α + \cot β + \cot γ$
C

$\tan α + \tan β + \tan γ = \tan α \tan β \tan γ$
D

$\tan αtan β + \tan βtan γ + \tan γtan α = 1$

Solution

$\tan (α + β) = \frac{15}{8}$
and $cosec γ = \frac{17}{8} \Rightarrow \tan γ = \frac{8}{15}$
$∴ α + β + γ = \frac{π}{2}$
Hence, (b) is true.
Also $\tan (α + β) = \frac{\tan α + \tan β}{1 - \tan αtan β}$
or $\cot γ = \frac{\tan α + \tan β}{1 - \tan αtan β}$
or $\tan α tanβ + tanβ \tan γ + \tan γ \tan α = 1$

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