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Basic Trigonometric Identities

If tanβ =2 ...

Question

If $t a n β = 2 s i n α s i n γ c o s e c (α + γ),$ then $c o t α, c o t β c o s t γ$ are in

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Solution

$t a n β = 2 s i n α s i n γ c o s e c (α + γ) s i n (α + γ) = 2 s i n α s i n γ c o t β (s i n α c o s γ + c o s α s i n α) = 2 s i n α s i n γ c o t β c o t γ + c o t α = 2 c o t β H e n c e c o t α, c o t β a n d c o t γ a r e i n A P$

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