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Question

Find tan22.5 using the half-angle formula.


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Solution

Find the value using half-angle formula

We know that tan(2x)=2tanx1-tan(x)2 and

Let y=tan(x) where x=22.5

So 2x=45

We know that tan(45)=1

So 2y1-y2=1

1-y2=2yy2+2y-1=0

We know if α and β are the roots of a quadratic equation of the form ax2+bx+c=0 then the roots are

α=-b+b2-4ac2aβ=-b-b2-4ac2a

Therefore the roots are 2-1 and -2-1

Since, tan22.5 lie in first quadrant so we will take the positive value.

Hence, the value of tan22.5 is 2-1


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