Given Expression: cos45/sec30 cosec30

To evaluate the given expression, we can follow these steps:

1. Evaluate cos45: The cosine of 45 degrees is equal to √2/2.

2. Evaluate sec30: The secant of 30 degrees is equal to 2.

3. Evaluate cosec30: The cosecant of 30 degrees is equal to 2.

4. Substitute the values obtained into the expression:
cos45/sec30 cosec30 = (√2/2) / (2 * 2).

Now, let's simplify the expression step by step:

cos45/sec30 cosec30 = (√2/2) / (4).

Since division is the same as multiplying by the reciprocal, we can rewrite the expression as:

cos45/sec30 cosec30 = (√2/2) * (1/4).

Simplifying the expression further:

cos45/sec30 cosec30 = √2/8.

Therefore, the final value of cos45/sec30 cosec30 is √2/8.

Explanation:

The given expression cos45/sec30 cosec30 involves trigonometric functions, which are used to relate angles and sides of a triangle. The cosine, secant, and cosecant functions are defined as follows:

- Cosine (cos): The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

- Secant (sec): The secant of an angle is defined as the reciprocal of the cosine function, i.e., secθ = 1/cosθ.

- Cosecant (cosec): The cosecant of an angle is defined as the reciprocal of the sine function, i.e., cosecθ = 1/sinθ.

In this case, we are given cos45 (the cosine of 45 degrees), sec30 (the secant of 30 degrees), and cosec30 (the cosecant of 30 degrees). By substituting these values into the given expression and simplifying it step by step, we find that the final value of cos45/sec30 cosec30 is √2/8.

This means that the expression evaluates to the ratio of the square root of 2 to 8, which can be further simplified if required.