Prove that
3sin6πsec3π−4sin65πcot4π=1
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Step-by-Step Solution
Step 1: Evaluate individual trigonometric values
First, we need to find the values of each trigonometric function in the given expression. We know that 6π radians is equal to 30∘, 3π radians is equal to 60∘, and 4π radians is equal to 45∘. For sin65π, we use the identity sin(π−x)=sinx.
Step 2: Substitute the values into the expression
Now, substitute the calculated trigonometric values back into the left-hand side of the given equation. This will allow us to simplify the expression.
Step 3: Simplify the expression
Perform the multiplications in each term. In the first term, 3×21×2 simplifies to 3×1=3. In the second term, 4×21×1 simplifies to 2×1=2.
Step 4: Calculate the final result
Finally, subtract the second term from the first term to get the final result. The result is 3−2=1. Since this matches the right-hand side of the original equation, the identity is proven.