# How do you write the expression as the sine, cosine, or tangent of the angle given cos45^circcos120^circ-sin45^circsin120^circ?

Aug 11, 2018

$\cos {165}^{\circ}$

#### Explanation:

$\text{using the "color(blue)"trigonometric identity}$

•color(white)(x)cos(x+y)=cosxcosy-sinxsiny

$\cos 45 \cos 120 - \sin 45 \sin 120 \text{ is the expansion of}$

$\cos \left(45 + 120\right) = \cos {165}^{\circ}$

Aug 11, 2018

#### Explanation:

The question is not clear.So the answer is given for angle ${165}^{\circ}$

We know that ,

color(red)(cosalphacosbeta-sinalphasinbeta=cos(alpha+beta)

Substitute , $\alpha = {45}^{\circ} \mathmr{and} \beta = {120}^{\circ}$

$\cos {45}^{\circ} \cos {120}^{\circ} - \sin {45}^{\circ} \sin {120}^{\circ}$=$\cos \left({45}^{\circ} + {120}^{\circ}\right)$=$\cos {165}^{\circ}$

:.cos165^circ=cos45^circ cos120^circ-sin45^circsin120^circ

Similarly , color(red)( sin(alpha+beta)=sinalphacosbeta+cosalphasinbeta

$\sin {165}^{\circ}$=$\sin \left({45}^{\circ} + {120}^{\circ}\right)$=sin45^circ cos120^circ+cos45^circsin120^circ

Now , color(red)(tan(alpha+beta)=(tanalpha+tanbeta)/(1- tanalphatanbeta)

$\tan {165}^{\circ} = \tan \left({45}^{\circ} + {120}^{\circ}\right) = \frac{\tan {45}^{\circ} + \tan {120}^{\circ}}{1 - \tan {45}^{\circ} \tan {120}^{\circ}}$