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How do you simplify tan(θ2) using the double angle identities?

Answer
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Hint: In order to simplify the given function using the double angle identity, first by using the relation between the sine, cosine and tangent function i.e. tanθ=sinθcosθ . Rewrite the given function in the form of sine and cosine function. Later using the double angles formula of sine, cosine function and solving the expression by substituting the values, we will get the exact values of tan(θ2) .
Formula used:
 tanθ=sinθcosθ
 sin2x=2sinxcosx
 cos2x=2cos2x1

Complete step-by-step answer:
We have given that,
 tan(θ2)
As we know that,
The relation between the trigonometric function of sine, cosine and tangent function is given by;
 tanθ=sinθcosθ
Therefore,
 tan(θ2)=sin(θ2)cos(θ2)
Using the double angle identities of sine and cosine function of trigonometry is given by;
 sin2x=2sinxcosx
And
 cos2x=2cos2x1
 sin(θ2)cos(θ2)=2sin(θ2)cos(θ2)2cos2(θ2)
Now,
Solving: 2cos2(θ2)
Using the double angle identity of cosine function is given by,
 cos2x=2cos2x1
In the given above question,
Let θ2=x
Therefore,
Applying the double angle identities;
Substituting θ2=x , we will get
 cos(2×(θ2))=2cos2(θ2)1
Adding 1 to both the sides,
 cos(2×(θ2))+1=2cos2(θ2)1+1
Simplifying the above, we will get
 1+cosθ=2cos2(θ2) ------- (1)
Solving: 2sin(θ2)cos(θ2)
Now,
We have,
 2sin(θ2)cos(θ2)
We can rewrite it as,
 2sin(θ2)cos(θ2)=sin(θ2)cos(θ2)+sin(θ2)cos(θ2)
As we know that,
 sin(A+B)=sinAcosB+cosAsinB
Applying this identity of sine function,
Here A=(θ2) , B=(θ2)
 2sin(θ2)cos(θ2)=sin(θ2)cos(θ2)+sin(θ2)cos(θ2)=sin((θ2)+(θ2))
 2sin(θ2)cos(θ2)=sin((θ2)+(θ2))
Solving the above,
 2sin(θ2)cos(θ2)=sin(2θ2)=sinθ
Therefore,
 2sin(θ2)cos(θ2)=sinθ ------- (2)
Substituting the value from equation (1) and equation (2), we will get
 tan(θ2)=sinθ1+cosθ
Hence, this is the required answer.
So, the correct answer is “ sinθ1+cosθ ”.

Note: While solving these types of questions, students must need to remember the formulas of double angle of trigonometric functions. One must be careful while noted down the values to avoid any error in the answer. They also need to consider the quadrant which is given in the question and they must remember which trigonometric function is positive or negative in all the four quadrants. We must know the basic concept of trigonometry and also the definition of the trigonometric function. The sine, cosine and the tangent are the three basic functions in introduction to trigonometry which shows the relation between all the sides of the triangles.

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