Question #12784

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

Expert's answer

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

(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

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