If cos ⁡ x = tan ⁡ y , cos ⁡ y = tan ⁡ z and cos ⁡ z = tan ⁡ x , then sin ⁡ x = 2 sin ⁡ θ where θ is (where, x , y , z , θ are acute angles)

Question

If cos ⁡ x = tan ⁡ y , cos ⁡ y = tan ⁡ z and cos ⁡ z = tan ⁡ x , then sin ⁡ x = 2 sin ⁡ θ where θ is (where, x , y , z , θ are acute angles), 15 o, 18 o, 22 1 2 o, 75 o

MARKS App · Accepted Answer

c o s 2 x = t a n 2 y = s e c 2 y - 1 = c o t 2 z - 1 So, 1 + c o s 2 x = c o t 2 z = c o s 2 z s i n 2 z = c o s 2 z 1 - c o s 2 z = t a n 2 x 1 - t a n 2 x = s i n 2 x c o s 2 x - s i n 2 x ⇒ 2 - s i n 2 x = s i n 2 x 1 - 2 s i n 2 x ⇒ 2 - s i n 2 x 1 - 2 s i n 2 x = s i n 2 x ⇒ 2 s i n 4 x - 6 s i n 2 x + 2 = 0 ⇒ s i n 2 x = 3 ± 9 - 4 2 = 3 ± 5 2 but s i n 2 x ∈ 0 , 1 so rejecting s i n 2 x = 3 + 5 2 we have, s i n 2 x = 3 - 5 2 = 5 - 1 2 2 ⇒ sin ⁡ x = 5 - 1 2 ⇒ 2 sin ⁡ θ = 5 - 1 2 ⇒ sin ⁡ θ = 5 - 1 4 ⇒ θ = 18 o

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