Use an appropriate half-angle formula to find the exact value of the expression. tan(15°)

Solution:

The relevant half angle formulas are:

Cos²θ = (1 + Cos2θ)/2         ---->(1)

Sin²θ =  (1- Cos2θ)/2         ---->(2)

Therefore,

Cos²15°  = (1 + Cos30°)/2 = (1 + √3/2)/2 

Sin²15°   = (1 - Cos30°)/2 = (1- √3/2)/2

Tan²15°  = Sin215° / Cos215°  = ((1 + √3/2)/2 ) / ((1- √3/2)/2 )

=(1 + √3/2)  / (1- √3/2)

Rationalising the above by multiplying and dividing  by (1 + √3/2), 

Tan²15° =   (1 + √3/2) / (1- √3/2)    ×     (1 + √3/2) / (1 + √3/2) 

= (1 + √3/2)² / ( 1- 3/4)

=4(1 + √3/2)²

Tan15° = √4(1 + √3/2)²

=   2(1 + √3/2)Answer

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Use an appropriate half-angle formula to find the exact value of the expression. tan(15°)

Summary:

Using an appropriate half-angle formula the exact value of the expression. tan(15°) is 2(1 + √3/2).