if (1+tan alpha)(1+ tan 4alpha) = 2, alpha belongs to (0, pi/16) then alpha is equal to
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Step-by-step explanation:
Given:-
(1+tan alpha)(1+ tan 4alpha) = 2, alpha belongs to (0, pi/16).
To find:-
The value of alpha?
Solution:-
(1+tan alpha)(1+ tan 4alpha) = 2
=>1+tan alpha +tan 4alpha +tan alpha tan 4alpha
=2
=>tan alpha +tan 4alpha +tan alpha tan 4alpha =2-1
=>tan alpha +tan 4alpha +tan alpha tan 4alpha
=1
=>tan alpha +tan 4alpha =1-tan alpha tan 4alpha
=>(tan alpha +tan 4alpha)/(1-tan alpha tan 4alpha )
=1
=>tan(alpha+4alpha)=1
=>tan 5alpha =1
=>tan 5 alpha=tan 45°
=>5 alpha =45°
=>alpha =45°/5
=>alpha =9°
=>alpha =180/20
=>alpha=π/20
Since alpha belongs to (0,π/16)
Therefore,alpha =9° or π/20
Used formulae:-
- Tan (A+B)=(TanA+TanB)/(1-TanATan B)
- π=180°
- Tan 45°=1
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