**Topic**

Calculate the value of the expression \(T = {\sin ^2}{25^0} + {\sin ^2}{75^0} + {\sin ^2}{115^0} + {\sin ^ 2}{165^0}\)

**Solution method – See details**

Step 1: Consider the relationship between interior angles *BILLION *with each other or with intermediate angles

Step 2: Convert the trigonometric values of the angles to the common trigonometric value of an angle

Step 3: Use the trigonometric formula \({\sin ^2}\alpha + {\cos ^2}\alpha = 1\) to simplify the expression *BILLION*

**Detailed explanation**

We have: \(\left\{ \begin{array}{l}\sin {25^0} = \cos ({90^0} – {25^0}) = \cos {65^0}\\ \sin {75^0} = \cos ({90^0} – {75^0}) = \cos {15^0}\\\sin {115^0} = \sin ({180^0} – {115^0}) = \sin {65^0}\\\sin {165^0} = \sin ({180^0} – {165^0}) = \sin {15^0}\end{ array} \right.\)

Then \(T = {\sin ^2}{25^0} + {\sin ^2}{75^0} + {\sin ^2}{115^0} + {\sin ^2}{165 ^0}\)\( = {\cos ^2}{65^0} + {\cos ^2}{15^0} + {\sin ^2}{65^0} + {\sin ^2} {15^0}\)

\( = ({\sin ^2}{65^0} + {\cos ^2}{65^0}) + ({\sin ^2}{15^0} + {\cos ^2}{15 ^0})\)\( = 1 + 1 = 2\)