Calculate the following trigonometric expression: E = cos x+2sin(x/2 +pi/12)sin(x/2 -pi/12)

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the value of E = cos x + 2*sin (x/2 + pi/12)*sin(x/2 -pi/12)

We use the formula 2 sin A sin B = - cos (A + B) + cos (A - B)

E = cos x + 2*sin (x/2 + pi/12)*sin(x/2 -pi/12)

=> cos x - cos (x/2 + pi/12 + x/2 - pi/12) + cos (x/2 + pi/12 - x/2 + pi/12)

=> cos x - cos (x/2 + x/2 ) + cos ( pi/12 + pi/12)

=> cos x - cos x + cos (pi/6)

=> cos (pi/6)

=> (sqrt 3) / 2

The required value of the expression is (sqrt 3)/2

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We notice that pi/12 is half angle of pi/6 = 30 degrees.

Then pi/12 = 15 degrees.

We'll use the identity:

sin a*sin b = (1/2)*[cos (a-b) - cos (a+b)]

sin(x/2 +pi/12)sin(x/2 -pi/12) = sin(x/2 + 15)sin(x/2 - 15)

2sin(x/2 + 15)sin(x/2 - 15) = cos(x/2 + 15 - x/2 + 15) - cos(x/2 + 15 + x/2 - 15)

2sin(x/2 + 15)sin(x/2 - 15) = cos(30) - cos(x) (1)

The product term will be substituted by (1):

E = cos x+ cos 30 - cos x

E = cos 30

E = sqrt3/2

The value of the given expression is E = sqrt3/2.

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