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What is equal to tan(π12)?
A. 23
B. 2+3
C. 23
D. 32

Answer
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Hint: To solve the question, we have to apply trigonometric identities and the values of trigonometric functions to arrive at the value of tan(π12).

Complete step-by-step answer:

We know that the formula for tan2α is given by 2tanα1tan2α
By substituting α=π12 in the above formula we get
tan(2π12)=2tan(π12)1(tan(π12))2
tan(π6)=2tan(π12)1(tan(π12))2 …….. (1)
We know that the value of tan(π6) is equal to 13
By substituting the above mentioned value in equation (1) we get,
13=2tan(π12)1(tan(π12))2
Cross multiply the above expression to obtain a quadratic expression of tan(π12).
1(tan(π12))2=3(2tan(π12))
(tan(π12))2+23(tan(π12))1=0 …….. (2)
We know that the solutions of the general quadratic expression ax2+bx+c=0 are given by b±b24ac2a
On comparing the above expression with equation (2) we get,
The values of a = 1, b = 23, c = -1
Thus, the possible values of tan(π12) are equal to 23±(23)24(1)(1)2(1)
We know that (ab)m=am×bm
=23±(22×3)+42
=23±12+42
=23±162
=23±42
Since we know that the value of 16=4×4=42=4.
tan(π12)=2(3±2)2
tan(π12)=3±2
We know that tanα is positive in the interval 0<α<π2 . Thus, we get
tan(π12)=23
The value of tan(π12) is equal to 23

Note: The possibility of mistake can be the calculation since the procedure of solving requires square root terms. The other possibility of mistake is not being able to choose the correct answer out of the obtained two values. The alternative way of solving can be, to calculate the value of cos(π12),sin(π12) since we know the value of cos(π6) is equal to 32 . By substituting the values in the formula cos2α=2cos2α1=12sin2α , we can calculate the value of cos(π12),sin(π12). Using the formula tanα=sinαcosα we can calculate the value of tan(π12). This method eases the procedure of solving.