How do you express #cos( (15 pi)/ 8 ) * cos (( 5 pi) /8 ) # without using products of trigonometric functions?

Answer 1

#cos((15pi)/8)cos ((5pi)/8)=1/2 cos((5pi)/2)+1/2 cos((5pi)/4)=-sqrt2/2#

#2cos A cos B=cos(A+B)+cos(A-B)#
#cosAcos B=1/2 (cos(A+B)+cos(A-B))#
#A=(15pi)/8, B=(5pi)/8#
#=>cos((15pi)/8)cos ((5pi)/8)=1/2 (cos((15pi)/8+(5pi)/8)+cos((15pi)/8-(5pi)/8))#
#=1/2 (cos((20pi)/8)+cos((10pi)/8)) #
#=1/2 cos((5pi)/2)+1/2 cos((5pi)/4) =0+ -sqrt2/2=-sqrt2/2 #
#cos((15pi)/8)cos ((5pi)/8)=1/2 cos((5pi)/2)+1/2 cos((5pi)/4)=-sqrt2/2#
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Answer 2

cos((15π)/8) * cos((5π)/8) can be expressed without using products of trigonometric functions by utilizing the sum-to-product identities. Specifically, we can use the identity cos(A) * cos(B) = (1/2) * [cos(A - B) + cos(A + B)].

Substituting A = (15π)/8 and B = (5π)/8 into this identity, we get:

cos((15π)/8) * cos((5π)/8) = (1/2) * [cos((15π)/8 - (5π)/8) + cos((15π)/8 + (5π)/8)].

Simplify the expressions inside the cosine functions:

(15π)/8 - (5π)/8 = (10π)/8 = (5π)/4

(15π)/8 + (5π)/8 = (20π)/8 = (5π)/2

Now, substitute these values back into the equation:

cos((15π)/8) * cos((5π)/8) = (1/2) * [cos((5π)/4) + cos((5π)/2)].

Using the values of cosine at (5π)/4 and (5π)/2:

cos((5π)/4) = -√2/2

cos((5π)/2) = 0

Substitute these values into the equation:

(1/2) * [-√2/2 + 0]

Simplify the expression:

= -(√2)/4.

Therefore, cos((15π)/8) * cos((5π)/8) = -(√2)/4.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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