Question

tan x = 1/5 and tan y = 1/239 prove that tan(4x-y)=1

Answer 1

Answer:

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Answer 2

Answer:

Hence tan(4x - y) = 1

Step-by-step explanation:

Given that

tan(x) = 1/5,  and tan(y) = 1/239

We have to prove that tan(4x - y) = 1

We know that, tan(x - y) = tan(x) - tan(y)/ 1 + tan(x)tan(y)

$tan(x -y) = \frac{tan(x) - tan(y)}{1+tan(x)tan(y)}$

tan(4x - y) = tan(4x) -tan(y)/1 + tan(4x)tan(y)

tan(2x) = 2tan(x)/1 - tan²(x) = 2(1/5)/1 - (1/5)²

$tan(2x) = \frac{2tan(x)}{1-tan^{2}(x) } = \frac{2\frac{1}{5} }{1-(\frac{1}{5})^{2} } = \frac{2}{5}*\frac{1}{1-\frac{1}{25} } = \frac{2}{5} *\frac{25}{24} =\frac{5}{12}$

$tan(4x) = \frac{2tan(2x)}{1-tan^{2}(2x) } =\frac{2*\frac{5}{12} }{1-(\frac{5}{12})^{2} } =\frac{5}{6} \frac{144}{144-25} =\frac{120}{119}$

$tan(4x-y) = \frac{\frac{120}{119}-\frac{1}{239} }{1+\frac{120}{119}*\frac{1}{235} } = \frac{(120)(239)-119}{(119)(239)+120} = \frac{28680-119}{28441+120} =\frac{28561}{28561} =1$

Therefore, tan(4x - y) = 1