Evaluate cos (5 𝜋 /12) + sin (5 𝜋 /12) / (cos 𝜋/ 12)-sin (𝜋 /12)

Question

It is two fractions over two fractions

Answer 1

Multiply top and bottom by the expression (cos(π/12) + sin(π/12)). This give us ...

[cos(5π/12)cos(π/12) + sin(5π/12)cos(π/12) + cos(5π/12)sin(π/12) + sin(5π/12)sin(π/12)]

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[cos²(π/12) - sin²(π/12)]

Then using sum and difference formulas for the numerator and the double-angle formula for cos for the denominator we get ...

[cos(π/3) + sin(π/2)] / [cos(π/6)] = 3/2 / √3/2 = √3

(The 1st and last terms in numerator become cos(π/3) while the middle two terms become sin(π/2).)

Answer 2

multiply the numerator and denominator by √2/2; then, as sin(π/4)=cos(π/4)=√2/2, substitute sin and cos of π/4 or -π/4 as appropriate, and we get

[sin(π/4)cos(5π/12)+cos(π/4)cos(5π/12)]/[cos(π/12)cos(π/4)-sin(π/12)sin(π/4)] =

sin(5π/12+π/4)/(cos(π/12+π/4) =

sin(2π/3)/cos(π/3) =

sin(π/3)/cos(π/3) =

tan(π/3) =

√3