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Calculus/pre cal 12th Circular functions II

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Question
prove the identities

sin (x+y)- sin (x-y) =2 cos x sin y


Find the exact value of cos 75

Answer
First you must know that :

sin (x+y) = sin(x) cos(y) + cos(x) sin(y)  and
sin (x-y) = sin(x) cos(y) - cos(x) sin(y).

Knowing the above two identities, we see that

sin (x+y)- sin (x-y)
= (  sin(x) cos(y) + cos(x) sin(y)) -  (sin(x) cos(y) - cos(x) sin(y))
= sin(x) cos(y) + cos(x) sin(y) - sin(x) cos(y) + cos(x) sin(y)
=          cos(x) sin(y)          + cos(x) sin(y)
= 2 cos(x) sin(y)

Now substitute x = 75  and y = 15,  we get

sin (75+15) - sin (75-15) = 2 cos ( 75) * sin (15)
that implies

sin (90)      - sin (60)      =  2 cos ( 75) * sin (15)

We know sin (90) and  sin (60) because both 60 and 90 are special angles.

But how about sin (15)?  There are two different ways:

I) We can use the power reducing formula that says:
2 sin^2 (15) = 1- cos( 30)

Since 30 is a special angle and we know  cos( 30) = sqrt(3)/2, therefore

sin(15) = sqrt( (1-  sqrt(3)/2 )/2  )

So now we have

   1         -  sqrt(3)/2      = 2 cos ( 75) * sqrt( (1-sqrt(3)/2 )/2)

I leave it to you to simplify and get solve for cos(75).

II)  75 and  15 are complementary because 75 and  15 together make a right angle.
So  sin (15) = cos ( 75) .

So we have from the equation:

  1         -  sqrt(3)/2      = 2 cos ( 75) *  cos ( 75)

So that  (cos(75))^2 = 1  -  sqrt(3)/2 )/2.

Solving for cos(75) by taking the positive square root yield your answer.  

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I can answer all calculus question. I am a Math Lecturer and I teach Math in a College, usually Calculus 1, Calculus 2 and Calculus 3 and Linear Algebra.

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