Solve cos(150^{circ})cos(210^{circ})-sin(150^{circ})sin(210^{circ})-{left(cos(15^circ)right)}^2+{left(sin(15^circ)right)}^2

$$\cos ( 150 ^ { \circ } ) \cos ( 210 ^ { \circ } ) - \sin ( 150 ^ { \circ } ) \sin ( 210 ^ { \circ } ) - { \left( \cos ( 15 ^ { \circ } ) \right) }^{ 2 } + { \left( \sin ( 15 ^ { \circ } ) \right) }^{ 2 }$$

Evaluate

$$0.1339745962155613160701050200652576$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\cos(150) \cos(210) - \sin(150) \sin(210) - 0.9659258262890683 ^ {2} + 0.25881904510252074 ^ {2}$$

Use \cos(x)\cos(y)=\frac{1}{2}\left(\cos(x-y)+\cos(x+y)\right) to obtain the result.

$$\frac{1}{2}\left(\cos(150-210)+\cos(150+210)\right)-\sin(150)\sin(210)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Subtract 210 from 150. Add 210 to 150.

$$\frac{1}{2}\left(\cos(-60)+\cos(360)\right)-\sin(150)\sin(210)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Use the property \cos(-x)=\cos(x).

$$\frac{1}{2}\left(\cos(60)+\cos(360)\right)-\sin(150)\sin(210)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Get the value of \cos(60) from trigonometric values table.

$$\frac{1}{2}\left(\frac{1}{2}+\cos(360)\right)-\sin(150)\sin(210)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Get the value of \cos(360) from trigonometric values table.

$$\frac{1}{2}\left(\frac{1}{2}+1\right)-\sin(150)\sin(210)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Do the calculations.

$$\frac{3}{4}-\sin(150)\sin(210)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Use \sin(x)\sin(y)=\frac{1}{2}\left(\cos(x-y)-\cos(x+y)\right) to obtain the result.

$$\frac{3}{4}-\frac{1}{2}\left(\cos(150-210)-\cos(150+210)\right)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Subtract 210 from 150. Add 210 to 150.

$$\frac{3}{4}-\frac{1}{2}\left(\cos(-60)-\cos(360)\right)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Use the property \cos(-x)=\cos(x).

$$\frac{3}{4}-\frac{1}{2}\left(\cos(60)-\cos(360)\right)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Get the value of \cos(60) from trigonometric values table.

$$\frac{3}{4}-\frac{1}{2}\left(\frac{1}{2}-\cos(360)\right)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Get the value of \cos(360) from trigonometric values table.

$$\frac{3}{4}-\frac{1}{2}\left(\frac{1}{2}-1\right)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Do the calculations.

$$\frac{3}{4}-\left(-\frac{1}{4}\right)-0.9659258262890683^{2}+0.25881904510252074^{2}$$

The opposite of -\frac{1}{4} is \frac{1}{4}.

$$\frac{3}{4}+\frac{1}{4}-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Add \frac{3}{4} and \frac{1}{4} to get 1.

$$1-0.9659258262890683^{2}+0.25881904510252074^{2}$$

Calculate 0.9659258262890683 to the power of 2 and get 0.93301270189221934897939208206489.

$$1-0.93301270189221934897939208206489+0.25881904510252074^{2}$$

Subtract 0.93301270189221934897939208206489 from 1 to get 0.06698729810778065102060791793511.

$$0.06698729810778065102060791793511+0.25881904510252074^{2}$$

Calculate 0.25881904510252074 to the power of 2 and get 0.0669872981077806650494971021301476.

$$0.06698729810778065102060791793511+0.0669872981077806650494971021301476$$

Add 0.06698729810778065102060791793511 and 0.0669872981077806650494971021301476 to get 0.1339745962155613160701050200652576.

$$0.1339745962155613160701050200652576$$