Cosec (90-51).sec 51-tan51.cot(90-51)
= sec51.sec51-tan51.tan51,
=sec^51-tan^51
=1

**Evaluation of cosec39. Sec51-tan51.cot39**

To evaluate the given expression, we will break it down step by step using the trigonometric identities and properties. Let's begin:

1. **Step 1: Finding the value of cosec39**
We know that cosec is the reciprocal of sine. So, we can evaluate cosec39 by finding the value of sine39 and taking its reciprocal.

- First, we need to convert the angle 39 degrees into radians. Since 1 radian is approximately equal to 57.3 degrees, we have: 39 degrees * (π/180) = 0.68 radians.
- Next, we find the value of sine39 using a calculator or trigonometric table. Let's assume the value of sine39 is x. Therefore, x = sin(0.68).
- Finally, we can evaluate cosec39 as the reciprocal of x: cosec39 = 1/x.

2. **Step 2: Finding the value of sec51**
Similarly, sec is the reciprocal of cosine. We can find sec51 by evaluating the value of cosine51 and taking its reciprocal.

- Convert the angle 51 degrees into radians: 51 degrees * (π/180) = 0.89 radians.
- Find the value of cosine51 using a calculator or trigonometric table. Let's assume the value of cosine51 is y. Therefore, y = cos(0.89).
- Evaluate sec51 as the reciprocal of y: sec51 = 1/y.

3. **Step 3: Finding the value of tan51**
Tan is the ratio of sine to cosine, so we can find tan51 by dividing the value of sine51 by the value of cosine51.

- Using a calculator or trigonometric table, find the value of sine51. Let's assume the value of sine51 is a. Therefore, a = sin(0.89).
- Find the value of tan51 as the ratio of a to y: tan51 = a/y.

4. **Step 4: Finding the value of cot39**
Cot is the reciprocal of tan, so we can find cot39 by evaluating the value of tan39 and taking its reciprocal.

- Convert the angle 39 degrees into radians: 39 degrees * (π/180) = 0.68 radians.
- Find the value of tan39 using a calculator or trigonometric table. Let's assume the value of tan39 is b. Therefore, b = tan(0.68).
- Evaluate cot39 as the reciprocal of b: cot39 = 1/b.

5. **Step 5: Evaluating the expression sec51 - tan51.cot39**
Now that we have the values of sec51, tan51, and cot39, we can substitute them into the expression and perform the arithmetic operations.

- Substitute the values: sec51 - tan51.cot39 = (1/y) - (a/y)(1/b).
- Simplify the expression by multiplying the terms: sec51 - tan51.cot39 = (1/y) - (a/b).
- Perform the subtraction: sec51 - tan51.cot39 = (1/y) - (a/b) = (b - a)/(by).

6. **Step