Find the value of sin2 5°+sin2 10°+sin2 15° + ................sin2 85° + sin2 90°

The given series is an arithmetic series of the squares of the sine of the angles in degrees. The common difference is 5°.

The sum of an arithmetic series is given by the formula:

S = n/2 * (a + l)

where: n is the number of terms, a is the first term, and l is the last term.

First, we need to find the number of terms (n). The first term is sin²5° and the last term is sin²90°. The common difference is 5°. So, we can find n by using the formula:

n = (l - a)/d + 1

where: l is the last term, a is the first term, and d is the common difference.

Substituting the given values, we get:

n = (90 - 5)/5 + 1 = 18

Now, we can find the sum of the series by substituting the values into the sum formula:

S = 18/2 * (sin²5° + sin²90°) = 9 * (sin²5° + 1)

Since sin²90° = 1 (as sin 90° = 1)

Now, we know that sin²θ = 1 - cos²θ. So, we can write sin²5° as 1 - cos²5°.

Substituting this into the sum formula, we get:

S = 9 * (1 - cos²5° + 1) = 9 * (2 - cos²5°)

This is the sum of the given series.