Solved on Dec 26, 2023
Prove that the exact value of the cosecant of 105 degrees is the square root of 6 minus the square root of 2.
STEP 1
Assumptions
1. We are working with trigonometric functions.
2. The angle provided is in degrees.
3. We will use trigonometric identities and formulas to find the exact value of .
4. We know that .
5. We will use the angle sum identity for sine: .
STEP 2
Express as the sum of two angles whose sine and cosine values we know exactly.
STEP 3
Write down the sine of using the angle sum identity.
STEP 4
Apply the angle sum identity for sine.
STEP 5
Substitute the exact values for , , , and .
STEP 6
Perform the multiplication for each term.
STEP 7
Simplify the multiplication.
STEP 8
Combine the terms over a common denominator.
STEP 9
Factor out from the numerator.
STEP 10
Now, find the cosecant of , which is the reciprocal of the sine.
STEP 11
Substitute the expression for .
STEP 12
Take the reciprocal of the fraction.
STEP 13
Multiply the numerator and denominator by the conjugate of the denominator to rationalize it.
STEP 14
Perform the multiplication in the numerator and use the difference of squares in the denominator.
STEP 15
Simplify the denominator.
STEP 16
Distribute the 4 in the numerator.
STEP 17
Divide each term by 2 to simplify further.
STEP 18
Recognize that and .
STEP 19
Simplify the square roots.
STEP 20
Simplify further to get the final result.
The exact value of is .
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