Solve -cos^2pi/6+1/2sin^2pi/4-1/2cosfrac{pi}{3}-tanfrac{pi}{4}

$$- \cos ^ { 2 } \frac { \pi } { 6 } + \frac { 1 } { 2 } \sin ^ { 2 } \frac { \pi } { 4 } - \frac { 1 } { 2 } \cos \frac { \pi } { 3 } - \tan \frac { \pi } { 4 }$$

Evaluate

$$-\frac{7}{4}=-1.75$$

Solution Steps

Get the value of \cos(\frac{\pi }{6}) from trigonometric values table.

$$-\left(\frac{\sqrt{3}}{2}\right)^{2}+\frac{1}{2}\left(\sin(\frac{\pi }{4})\right)^{2}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$-\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\frac{1}{2}\left(\sin(\frac{\pi }{4})\right)^{2}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

Get the value of \sin(\frac{\pi }{4}) from trigonometric values table.

$$-\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\frac{1}{2}\times \left(\frac{\sqrt{2}}{2}\right)^{2}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$-\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\frac{1}{2}\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

Multiply \frac{1}{2} times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2\times 2^{2}}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2^{2} and 2\times 2^{2} is 2\times 2^{2}. Multiply -\frac{\left(\sqrt{3}\right)^{2}}{2^{2}} times \frac{2}{2}.

$$-\frac{2\left(\sqrt{3}\right)^{2}}{2\times 2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2\times 2^{2}}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

Since -\frac{2\left(\sqrt{3}\right)^{2}}{2\times 2^{2}} and \frac{\left(\sqrt{2}\right)^{2}}{2\times 2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}}{2\times 2^{2}}-\frac{1}{2}\cos(\frac{\pi }{3})-\tan(\frac{\pi }{4})$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$\frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}}{2\times 2^{2}}-\frac{1}{2}\times \frac{1}{2}-\tan(\frac{\pi }{4})$$

Multiply \frac{1}{2} and \frac{1}{2} to get \frac{1}{4}.

$$\frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}}{2\times 2^{2}}-\frac{1}{4}-\tan(\frac{\pi }{4})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 2^{2} and 4 is 8. Multiply \frac{1}{4} times \frac{2}{2}.

$$\frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}}{8}-\frac{2}{8}-\tan(\frac{\pi }{4})$$

Since \frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}}{8} and \frac{2}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}-2}{8}-\tan(\frac{\pi }{4})$$

Get the value of \tan(\frac{\pi }{4}) from trigonometric values table.

$$\frac{-2\left(\sqrt{3}\right)^{2}+\left(\sqrt{2}\right)^{2}-2}{8}-1$$

The square of \sqrt{3} is 3.

$$\frac{-2\times 3+\left(\sqrt{2}\right)^{2}-2}{8}-1$$

Multiply -2 and 3 to get -6.

$$\frac{-6+\left(\sqrt{2}\right)^{2}-2}{8}-1$$

The square of \sqrt{2} is 2.

$$\frac{-6+2-2}{8}-1$$

Add -6 and 2 to get -4.

$$\frac{-4-2}{8}-1$$

Subtract 2 from -4 to get -6.

$$\frac{-6}{8}-1$$

Reduce the fraction \frac{-6}{8} to lowest terms by extracting and canceling out 2.

$$-\frac{3}{4}-1$$

Subtract 1 from -\frac{3}{4} to get -\frac{7}{4}.

$$-\frac{7}{4}$$

Factor

$$-\frac{7}{4} = -1\frac{3}{4} = -1.75$$