Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help

cos (-5pi/12)

That is -75 degrees. Use the formula for cos (A + B), where A = -45 degrees and B = -30 degrees.

cos (-5pi/12) = cos (5pi/12)
= cos45deg*cos30deg - sin45deg*sin30deg
= (sqrt2)/2*(sqrt3)/2 - sqrt2/4
= [(sqrt6)- (sqrt2)]/4
= sqrt2*(sqrt3 -1)/4

To evaluate cos (-5π/12) without using a calculator, we can use the cosine angle sum and difference identities.

Recall the cosine angle difference identity:
cos(A - B) = cos A cos B + sin A sin B

In this case, we have -5π/12 as our angle, which can be expressed as the difference of two angles: -π/3 - π/4.

cos (-5π/12) = cos((-π/3) - (π/4))

Using the cosine angle difference identity, we have:
cos((-π/3) - (π/4)) = cos(-π/3) cos(π/4) + sin(-π/3) sin(π/4)

Now, we need to evaluate cos(-π/3) and sin(-π/3), as well as cos(π/4) and sin(π/4).

To find cos(-π/3):
Since the cosine function has a period of 2π, we can shift the angle by adding 2π and still get the same value.
So, -π/3 + 2π = -π/3 + 6π/3 = 5π/3.
Since the cosine function is an even function, cos(-π/3) = cos(5π/3).

To find sin(-π/3):
Since the sine function is an odd function, sin(-π/3) = -sin(π/3).

To find cos(π/4) and sin(π/4):
These are special angles that can be found using the unit circle. At π/4, the coordinates of a point on the unit circle are (cos(π/4), sin(π/4)), which is (√2/2, √2/2).

Now, let's substitute these values into our equation:

cos((-π/3) - (π/4)) = cos(-π/3) cos(π/4) + sin(-π/3) sin(π/4)
= cos(5π/3) * (√2/2) + (-sin(π/3)) * (√2/2)

Finally, simplify the expression:

cos(5π/3) * (√2/2) + (-sin(π/3)) * (√2/2) = (√2)/2 * (cos(5π/3) - sin(π/3))

Now, we need to find the values of cos(5π/3) and sin(π/3) using the unit circle or other trigonometric identities.

From the unit circle, we know that cos(5π/3) = -1/2 and sin(π/3) = √3/2.

Substituting these values into the expression:

(√2)/2 * (cos(5π/3) - sin(π/3)) = (√2)/2 * (-1/2 - √3/2)
= -√2/4 - √6/4

Therefore, cos(-5π/12) = -√2/4 - √6/4 when evaluated exactly.