Answer to Question #12784 in Trigonometry for Achini Nawarathna

Question

if tan(x+π/12)cot(x+ π/12)=λ proof that sin2x-(λ+1)/2(λ-1)......(I PROVED IT)
and then prove that tan(x+π/12)cot(x+ π/12) is not between 1/3 and 3 for any x

Answer 1

tan(x+π/12)cot(x-π/12)=λ
(sin(x+π/12)cos(x-π/12 )/(sin(x-π/12)cos(x+π/12) ) = λ
remember
sin a cos b = (1/2) [sin (a+b) +sin(a-b)]
(1/2) [sin (2x) + sin(π/6)] /{(1/2)[ sin(2x) - sin(π/6) } = λ
[sin (2x) + 1/2] /[sin(2x) - 1/2] = λ
sin(2x) = z
(z + 1/2)/(z - 1/2) = λ
(2z + 1)/(2z - 1) = λ
2z + 1 = λ(2z - 1)
2z + 1 = 2λz - λ
2z - 2λz = - λ - 1
2z(λ - 1) = λ + 1
z = (λ + 1)/(2 (λ - 1))
sin (2x) = (λ + 1)/(2 (λ - 1))

second part
for any x- 1 ≤ sin (2x) ≤ 1
therefore
(λ + 1)/(2 (λ - 1)) ≤ 1
(λ + 1)/(2 (λ - 1)) - 1 ≤ 0
(λ + 1 - 2 λ + 2)/(2 (λ - 1)) ≤ 0
(3 - λ)/(2(λ - 1)) ≤ 0
λ ≤1 OR λ ≥ 3
AND
(λ + 1)/(2 (λ - 1)) ≥ - 1
(λ + 1)/(2 (λ - 1)) + 1 ≥ 0
(λ + 1 + 2λ - 2)/(2 (λ - 1)) ≥ 0
(3λ - 1)/(2 (λ - 1)) ≥ 0
λ ≤ 1/3 OR λ ≥ 1
(λ ≤1 OR λ ≥ 3) AND (λ ≤ 1/3 OR λ ≥ 1)
means
λ ≤ 1/3 OR λ ≥ 3
tan(x+π/12)cot(x-π/12)=λ is not between 1/3 and 3