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Find tan22.5 using a half- angle formula.

Answer
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Hint: In this problem, we have to find the value of tan22.5 using a half -angle formula. We will use the half angle formula for tangent as follows.
 tan2θ=2tanθ1tan2θ
We will put θ=22.5 in a half angle formula and then solve by cross multiplying to get the desired value. We will also use the value of tan45=1 .

Complete step by step solution:
This question is based on application of trigonometric formulas. Trigonometric formula is based on the relationship between T-ratio of angles, identify sides etc.
Half angle formula is based on the formula for the sum of two angles.
For example, tan(A+B)=tanA+tanB1tanAtanB
If we put A=B then the formula becomes
 tan2A=2tanA1tan2A
Considering the given question, we have to find the value of tan22.5 .
From half angle formula we have,
 tan2θ=2tanθ1tan2θ
 Let, θ=22.5 Then 2θ=45 .
Putting θ=22.5 in above half angle formula we get
 tan45=2tan22.51tan222.5
We know that tan45=1
Hence we have,
  2tan22.51tan222.5=1
On cross multiplication we have,
  2tan22.5=1tan222.5
Adding tan222.51 to both sides, we have,
 tan222.5+2tan22.51=0
Let , x=tan22.5 then we have,
 x2+2x1=0
This is a quadratic equation. We know that the solution of quadratic equation ax2+bx+c=0 is given by x=b±b24ac2a .
Here, a=1 , b=2 and c=1 .
Hence , x=2±224×1×(1)2×1=2±222
Hence , x=1±2 .
Therefore, tan22.5=1+2 or tan22.5=12
Since , θ=22.5 lies in the first quadrant .
Therefore the value of tan22.5 is positive.
Hence, tan22.5=1+2
Hence the value of tan22.5 is 21
So, the correct answer is “ 21 ”.

Note: Values of all T-ratios are positive in the first quadrant. While In second quadrant, only the value of sine is positive. In the third quadrant, only tangent is positive and in the fourth quadrant, only cosine is positive.
Quadratic equations can also be solved by splitting the middle term such that sum of terms is middle term and product is constant term.
Some important trigonometry half angle formula are
 sin2θ=2sinθcosθ
 cos2θ=cos2θsin2θ