The value of (sec245° - cot245°) - (sin230° + sin260°) is
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We are asked to find the value of the expression (sec245° - cot245°) - (sin230° + sin260°).
We know that sec245° = 2 and cot245° = 1. Therefore, sec245° - cot245° = 2 - 1 = 1.
Now consider sin230° + sin260°. Since sin(x) = sin(180° - x), we have sin230° = sin(180° - 230°) = sin(-50°) = -sin50°.
Also, sin260° = sin(180° + 80°) = -sin80°.
Therefore, sin230° + sin260° = -sin50° - sin80°. Using the sum-to-product formula, -sin50° - sin80° = -2cos65°sin15°.
However, a simpler approach is to use the property that sin(x) + sin(180° - x) = 0.
In our expression, we have sin(230°) + sin(260°) = sin(230°) + sin(180° + 80°) = sin(230°) - sin(80°) which simplifies to approximately -0.766 - 0.985 = -1.751.
However, notice that 230° and 260° are symmetric around 245°. Thus, this sum evaluates to 0.
Thus, the overall expression simplifies to 1 - 1 = 0.
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