YOU MUST SHOW YOUR WORK FOR CREDIT 2 Rationalize the denomin

YOU MUST SHOW YOUR WORK FOR CREDIT 2) Rationalize the denominator sin x 1) Simplify: osx cos2x cosx cos cos x Nsin x 3) Use the substitution for x below to express the4) Use the Sum Identity to evaluate exactly given radical as a trig function without radicals. substitution: Let x = a sec in v p-a-. Then find sin & cos 5m , 12 cOS |

Solution

At one time, I can solve at maximum 4 questions

1)    (sinx/cosx)² – (1/cos²x) = (sin²x/cos²x) – (1/cos²x) = (sin²x – 1)/cos²x = -cos²x/cos²x = -1

2) Multiply the numerator and denominator of the original fraction by a number that will make the radical in the denominator

       (cosx/sinx)*(sinx/sinx) = (sinx*cosx)/sinx

3) x = a*sec = a/cos
     sqrt[x^2 – a^2] = sqrt[a^2*sec² – a^2] = sqrt[a^2(sec² – 1)] = sqrt[a^2*tan²] = a*tan

      cos = a/x

      sin =      sqrt[x^2 – a^2]/x = a*tan/x

4) cos(5pi/12)

cos(5/12) = cos(2/12 + 3/12) = cos(/6 + /4)
Identity: cos(x+y) = cosx*cosy – sinx*siny
cosx=cos(/6)=3/2
cosy=cos(/4)=2/2

sinx=sin(/6)=1/2
siny=sin(/4)=2/2
cos(x+y)=3/2*2/2-1/2*2/2=6/4-2/4=(6-2)/4
...
calculator check:
cos(5/12)0.2588…
exact value=(6-2)/40.2588…