Question

if (1+tan alpha)(1+ tan 4alpha) = 2, alpha belongs to (0, pi/16) then alpha is equal to​

Answer 1

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