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Trigonometry: Evaluating the Expression 1 - 4cos^2(x - 5π/12)
In this article, we will evaluate the trigonometric expression 1 - 4cos^2(x - 5π/12). To do this, we will use the properties of trigonometric functions and algebraic manipulations.
Step 1: Simplify the Expression
First, let's rewrite the expression by using the cosine identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In our case, A = x and B = 5π/12. Then:
cos(x - 5π/12) = cos(x)cos(5π/12) + sin(x)sin(5π/12)
Now, we can rewrite the original expression as:
1 - 4(cos(x)cos(5π/12) + sin(x)sin(5π/12))^2
Step 2: Expand the Squared Term
Using the binomial theorem, we expand the squared term:
(cos(x)cos(5π/12) + sin(x)sin(5π/12))^2 = cos^2(x)cos^2(5π/12) + 2cos(x)sin(x)cos(5π/12)sin(5π/12) + sin^2(x)sin^2(5π/12)
Now, we can rewrite the expression as:
1 - 4(cos^2(x)cos^2(5π/12) + 2cos(x)sin(x)cos(5π/12)sin(5π/12) + sin^2(x)sin^2(5π/12))
Step 3: Simplify the Expression
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify the expression:
1 - 4(cos^2(x)cos^2(5π/12) + 2cos(x)sin(x)cos(5π/12)sin(5π/12) + (1 - cos^2(x))sin^2(5π/12))
Now, we can combine like terms:
= 1 - 4cos^2(x)cos^2(5π/12) - 8cos(x)sin(x)cos(5π/12)sin(5π/12) + 4sin^2(x)sin^2(5π/12)
Final Result
The final result is a simplified expression:
1 - 4cos^2(x - 5π/12) = 4sin^2(x)sin^2(5π/12) - 8cos(x)sin(x)cos(5π/12)sin(5π/12) - 4cos^2(x)cos^2(5π/12) + 1
This expression has been simplified using trigonometric identities and algebraic manipulations.
