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Question
Select the correct option from the given alternatives:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
Options
`17/7`
`-17/7`
`7/17`
`-7/17`
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Solution
`"tan"(2"tan"^-1 (1/5) - pi/4) = \underline(-7/17)`.
Explanation:
`tantheta = 1/5`
`tan(2theta) = (2tantheta)/(1-tan^2theta)`
`tan(2theta) = 2(1/5)/1-(1/5)^2 = (2/5)/(1-1/25) = (2/5)/(24/25) = 2/5 xx 25/24 = 50/120 = 5/12`
`tan(2theta) = 5/12`
`tan(2theta-pi/4) = (tan(2theta)-tan pi/4)/(1+tan(2theta)tan pi/4)`
`tan(2theta - pi/4) = (5/12 -1)/(1+5/12 xx 1) = ((5-12)/12)/(1+5/12) = (-7/12)/(17/12) = -7/17`
`=-7/17`
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