We know that ,

i ) cos ( 90 - A ) = sin A 

ii ) sin C - Sin D = 2cos( C + D )/2 X sin(C - D )/2

Cos 10 - sin 10 

= cos (90-80) - sin10

= sin80 - sin10

= 2cos(80+10)/2 X sin(80-10)/2

= 2cos45sin35

= 2 X ( 1/sqrt2) sin35

= ( sqrt2 X sqrt2 )/ ( sqrt2 ) X sin35

= ( sqrt 2) X sin35

The value of cos10-sin10 can be calculated using trigonometric identities and the unit circle. Let's break down the steps to find the value of cos10-sin10:

Step 1: Understanding the Trigonometric Functions
Trigonometric functions are mathematical functions that relate angles to ratios of side lengths in a right triangle. The two primary trigonometric functions used in this problem are cosine (cos) and sine (sin).

Step 2: Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables involved. One important identity is the Pythagorean identity, which states that sin²θ + cos²θ = 1. This identity connects the values of sine and cosine for any angle θ.

Step 3: The Unit Circle
The unit circle is a circle with a radius of 1 unit. It is used to define the values of sine and cosine for specific angles. The x-coordinate of a point on the unit circle represents the cosine of the angle, while the y-coordinate represents the sine of the angle.

Step 4: Finding cos10 and sin10
To find the values of cos10 and sin10, we can refer to the unit circle. The angle 10° is in the first quadrant, and its reference angle (the acute angle formed between the terminal side of the angle and the x-axis) is 10°.

On the unit circle, the coordinates for the point corresponding to the reference angle of 10° are approximately (0.9848, 0.1736). Therefore, cos10 ≈ 0.9848 and sin10 ≈ 0.1736.

Step 5: Calculating cos10-sin10
Now that we have the values of cos10 and sin10, we can substitute them into the expression cos10-sin10.

cos10 - sin10 = 0.9848 - 0.1736 ≈ 0.8112

Therefore, the value of cos10 - sin10 is approximately 0.8112.