Question

cos36-sin18=1/2 (prove it)

Answer 1

Answer:

How can I prove that cos 36° -sin 18° =1/2?

We have the identity cos (π/2 -x) = sin x. Using that, we can write cos (36) = cos (90 - 54) = sin (54)

Now, we try to find the proof of the new equation

sin(54) - sin (18) = ?

Using the product sum identity,

sin(x) - sin(y) = 2*cos((x+y)/2)*sin((x-y)/2),

we have for x=54 and y=18:

sin(54)−sin(18)=2∗cos(36)∗sin(18)

Using the double angle formula,

sin(2t) = 2*sin(t)*cos(t),

we have for t=18:

sin(36) = 2*sin(18)cos(18), therefore

sin(18)=sin(36)(2∗cos(18))

Substituting back into the earlier equation in step 1 and simplifying:

sin(54)−sin(18)=cos(36)∗sin(36)cos(18)

Using the complement of the trig function,

cos(t) = sin(90 - t),

we have for t=18:

sin(54)−sin(18)=cos(36)∗sin(36)sin(72)

Reapplying the double angle formula and substituting, with t=36, we have:

sin(54)−sin(18)=cos(36)∗sin(36)2∗sin(36)cos(36)

Simplifying:

sin(54)−sin(18)=12

i.e,cos(36)−sin(18)=12

Hope that helps.