We can see that the given triangle is a right triangle because 652=162+632. Therefore, angle C is a right angle. Now we use the following trigonometric identities: cos‌2‌A=2cos2A−1 cos‌2‌B=2cos2B−1 cos‌2‌C=2cos2C−1 Now we know that, cos‌C=cos‌90∘=0. Therefore, cos‌2‌C=2cos2C−1=2×02−1=−1. Hence, we have cos‌2‌A+cos‌2‌B+cos‌2‌C=(2cos2A−1)+(2cos2B−1)+(−1)=2(cos2A+cos2B)−3. Now, we use the identity cos2A+cos2B=1+cos‌A‌cos‌B (which can be derived by using the angle subtraction formula for cosine: cos(A−B)=cos‌A‌cos‌B+sin‌Asin‌B and then squaring both sides and using the identity sin‌2x+cos2x=1 ). Therefore, cos‌2‌A+cos‌2‌B+cos‌2‌C=2(1+cos‌A‌cos‌B)−3=2‌cos‌A‌cos‌B−1. Now, we can find cos‌A and cos‌B using the definition of cosine: cos‌A=‌
AB
AC
=‌
16
65
‌ and ‌cos‌B=‌
BC
AC
=‌
63
65
‌. ‌ So, we get cos‌2‌A+cos‌2‌B+cos‌2‌C=2×‌