Sum of Trigonometric Ratios in Terms of Their Product
If α+β=γ an...
Question
If α+β=γ and tanγ=22, a is the arithmetic and b is the geometric mean respectively between tanα and tanβ, then the value of a3(1−b2)3 is equal to
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Solution
Given tanα,b,tanβ are in GP ⇒b2=tanαtanβ ....(1) Also given tanα,a,tanβ are in AP ⇒tanα+tanβ=2a ...(2) Also, given γ=α+β tanγ=tan(α+β) tanγ=tanα+tanβ1−tanαtanβ ⇒22=2a1−b2 (by (1) and (2)) ⇒a3(1−b2)3=113=1331.