How do you evaluate cos315?

Hint:The above question is based on trigonometric functions which shows the relationship between the angles and sides of the triangle. The expression $\cos 135$ is a cosine trigonometric function which can be solved by applying trigonometric identities by knowing the behavior of a function in the quadrant.

Complete step by step solution:
Given, is the angle which is 315. So, to solve the value of functions we need to to know the
trigonometric identities of cosine function. Also, cosine function sign in very co-ordinate also matters to solve the above function.

The sign of trigonometric function depends on the signs of x and y coordinate i.e., Depends on which quadrant the function is lying.

The sign of all the six trigonometric functions in the first quadrant is positive since x and y coordinates are both positive. In the second quadrant sine and cosecant are positive and in third only tangent and cotangent are positive. The fourth quadrant has only cosine and secant are positive.

In the question above we need to evaluate every term of cosine function using some trigonometric
Identities.

The cosine function has the identity \[\cos (360n - \theta ) = \cos \theta \]. Since it lies in the fourth the quadrant which has the sign of cosine function is positive.
\[\cos ({360^ \circ } - {45^ \circ }) = \cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\]

Therefore ,we get the value as \[\dfrac{1}{{\sqrt 2 }}\]

Note: An important thing to note is that the identity \[\cos (360n - \theta ) = \cos \theta \] when applied to the cosine function gives n value as 1 as n is a positive integer. So, when n=1, then \[360 \times 1 - 45\] will give the value as 315.