# Answer to Question #12784 in Trigonometry for Achini Nawarathna

Question #12784
if tan(x+&pi;/12)cot(x+ &pi;/12)=&lambda; proof that sin2x-(&lambda;+1)/2(&lambda;-1)......(I PROVED IT) and then prove that tan(x+&pi;/12)cot(x+ &pi;/12) is not between 1/3 and 3 for any x
1
2012-08-14T09:08:29-0400
tan(x+&pi;/12)cot(x-&pi;/12)=&lambda;
(sin(x+&pi;/12)cos(x-&pi;/12 )/(sin(x-&pi;/12)cos(x+&pi;/12) ) = &lambda;
remember
sin a cos b = (1/2) [sin (a+b) +sin(a-b)]
(1/2) [sin (2x) + sin(&pi;/6)] /{(1/2)[ sin(2x) - sin(&pi;/6) } = &lambda;
[sin (2x) + 1/2] /[sin(2x) - 1/2] = &lambda;
sin(2x) = z
(z + 1/2)/(z - 1/2) = &lambda;
(2z + 1)/(2z - 1) = &lambda;
2z + 1 = &lambda;(2z - 1)
2z + 1 = 2&lambda;z - &lambda;
2z - 2&lambda;z = - &lambda; - 1
2z(&lambda; - 1) = &lambda; + 1
z = (&lambda; + 1)/(2 (&lambda; - 1))
sin (2x) = (&lambda; + 1)/(2 (&lambda; - 1))

second part
for any x- 1 &le; sin (2x) &le; 1
therefore
(&lambda; + 1)/(2 (&lambda; - 1)) &le; 1
(&lambda; + 1)/(2 (&lambda; - 1)) - 1 &le; 0
(&lambda; + 1 - 2 &lambda; + 2)/(2 (&lambda; - 1)) &le; 0
(3 - &lambda;)/(2(&lambda; - 1)) &le; 0
&lambda; &le;1 OR &lambda; &ge; 3
AND
(&lambda; + 1)/(2 (&lambda; - 1)) &ge; - 1
(&lambda; + 1)/(2 (&lambda; - 1)) + 1 &ge; 0
(&lambda; + 1 + 2&lambda; - 2)/(2 (&lambda; - 1)) &ge; 0
(3&lambda; - 1)/(2 (&lambda; - 1)) &ge; 0
&lambda; &le; 1/3 OR &lambda; &ge; 1
(&lambda; &le;1 OR &lambda; &ge; 3) AND (&lambda; &le; 1/3 OR &lambda; &ge; 1)
means
&lambda; &le; 1/3 OR &lambda; &ge; 3
tan(x+&pi;/12)cot(x-&pi;/12)=&lambda; is not between 1/3 and 3

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