SOLUTION: Use identities to write the expression as a function of x. tan(x+pi/6)

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Question 1204959: Use identities to write the expression as a function of x.
tan(x+pi/6)
Found 3 solutions by MathLover1, ikleyn, math_tutor2020:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!


use identities:


...,



then we have



since and









Answer by ikleyn(52910)   (Show Source): You can put this solution on YOUR website!
.
Use identities to write the expression as a function of x.
tan(x+pi/6)
~~~~~~~~~~~~~~~~~~~ 

Use trigonometric identity - the addition formula for tangents

    tan(a+b) = .    (1)


It gives

     = .    (2)


Next, use the formula of half-angle for tangents to express  

     =  =  =  =  = .    (3)


Continue (2)

     =  =  = .


ANSWER.    simplified is  .

Solved



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

The other tutors mathlover and ikleyn have great answers.
Despite their final answers looking completely different, they are actually equivalent.

To go from mathlover's answer to ikleyn's answer, multiply top and bottom by


in the numerator becomes
where




in the denominator becomes



Therefore we can claim that
is an identity.
A graphing tool can be used to confirm that the two sides are the same.

Desmos Graph
https://www.desmos.com/calculator/av7ncyhcj5
Note how one curve perfectly overlaps the other.

It is a true equation for all x in the domain.
That domain being x is the set of all real numbers but where n is an integer.
If was the case, then it causes a division by zero error.
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