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To evaluate the given expression [tex]\(\cos^2 60^\circ + \sec^2 150^\circ - \csc^2 210^\circ\)[/tex], we will break it down into its components and calculate each term individually.
### Step 1: Calculate [tex]\(\cos^2 60^\circ\)[/tex]
The cosine of 60 degrees is given by:
[tex]\[ \cos 60^\circ = \frac{1}{2} \][/tex]
Therefore:
[tex]\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \][/tex]
### Step 2: Calculate [tex]\(\sec^2 150^\circ\)[/tex]
The secant function is the reciprocal of the cosine function. First, we find [tex]\(\cos 150^\circ\)[/tex]:
[tex]\[ \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2} \][/tex]
Thus, the secant is:
[tex]\[ \sec 150^\circ = \frac{1}{\cos 150^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \][/tex]
And the square of the secant is:
[tex]\[ \sec^2 150^\circ = \left(-\frac{2\sqrt{3}}{3}\right)^2 = \frac{4 \cdot 3}{9} = \frac{12}{9} = \frac{4}{3} \][/tex]
### Step 3: Calculate [tex]\(\csc^2 210^\circ\)[/tex]
The cosecant function is the reciprocal of the sine function. First, we find [tex]\(\sin 210^\circ\)[/tex]:
[tex]\[ \sin 210^\circ = -\sin 30^\circ = -\frac{1}{2} \][/tex]
Thus, the cosecant is:
[tex]\[ \csc 210^\circ = \frac{1}{\sin 210^\circ} = \frac{1}{-\frac{1}{2}} = -2 \][/tex]
And the square of the cosecant is:
[tex]\[ \csc^2 210^\circ = (-2)^2 = 4 \][/tex]
### Step 4: Combine the values
Now that we have all the individual values, we can substitute them into the original expression:
[tex]\[ \cos^2 60^\circ + \sec^2 150^\circ - \csc^2 210^\circ = \frac{1}{4} + \frac{4}{3} - 4 \][/tex]
To combine these terms, we need a common denominator. The common denominator for 4, 3, and 1 is 12. Therefore:
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \frac{4}{3} = \frac{16}{12} \][/tex]
[tex]\[ 4 = \frac{48}{12} \][/tex]
Adding and subtracting these fractions:
[tex]\[ \frac{3}{12} + \frac{16}{12} - \frac{48}{12} = \frac{3 + 16 - 48}{12} = \frac{-29}{12} \][/tex]
Hence, the result of evaluating the expression [tex]\(\cos^2 60^\circ + \sec^2 150^\circ - \csc^2 210^\circ\)[/tex] is:
[tex]\[ -2.416666666666665 \approx -2.42 \][/tex]
Since none of the choices match this result exactly, the correct assessment is that no given options are correct for this problem.
### Step 1: Calculate [tex]\(\cos^2 60^\circ\)[/tex]
The cosine of 60 degrees is given by:
[tex]\[ \cos 60^\circ = \frac{1}{2} \][/tex]
Therefore:
[tex]\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \][/tex]
### Step 2: Calculate [tex]\(\sec^2 150^\circ\)[/tex]
The secant function is the reciprocal of the cosine function. First, we find [tex]\(\cos 150^\circ\)[/tex]:
[tex]\[ \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2} \][/tex]
Thus, the secant is:
[tex]\[ \sec 150^\circ = \frac{1}{\cos 150^\circ} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \][/tex]
And the square of the secant is:
[tex]\[ \sec^2 150^\circ = \left(-\frac{2\sqrt{3}}{3}\right)^2 = \frac{4 \cdot 3}{9} = \frac{12}{9} = \frac{4}{3} \][/tex]
### Step 3: Calculate [tex]\(\csc^2 210^\circ\)[/tex]
The cosecant function is the reciprocal of the sine function. First, we find [tex]\(\sin 210^\circ\)[/tex]:
[tex]\[ \sin 210^\circ = -\sin 30^\circ = -\frac{1}{2} \][/tex]
Thus, the cosecant is:
[tex]\[ \csc 210^\circ = \frac{1}{\sin 210^\circ} = \frac{1}{-\frac{1}{2}} = -2 \][/tex]
And the square of the cosecant is:
[tex]\[ \csc^2 210^\circ = (-2)^2 = 4 \][/tex]
### Step 4: Combine the values
Now that we have all the individual values, we can substitute them into the original expression:
[tex]\[ \cos^2 60^\circ + \sec^2 150^\circ - \csc^2 210^\circ = \frac{1}{4} + \frac{4}{3} - 4 \][/tex]
To combine these terms, we need a common denominator. The common denominator for 4, 3, and 1 is 12. Therefore:
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
[tex]\[ \frac{4}{3} = \frac{16}{12} \][/tex]
[tex]\[ 4 = \frac{48}{12} \][/tex]
Adding and subtracting these fractions:
[tex]\[ \frac{3}{12} + \frac{16}{12} - \frac{48}{12} = \frac{3 + 16 - 48}{12} = \frac{-29}{12} \][/tex]
Hence, the result of evaluating the expression [tex]\(\cos^2 60^\circ + \sec^2 150^\circ - \csc^2 210^\circ\)[/tex] is:
[tex]\[ -2.416666666666665 \approx -2.42 \][/tex]
Since none of the choices match this result exactly, the correct assessment is that no given options are correct for this problem.
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