 Hi, I am Meenu and going to help you to solve the following problem. It says prove the following We have to prove that cos 6x is equal to 32 cos to the power 6x minus 48 cos to the power 4x plus 18 cos square x minus 1 So let us first understand the formulae which will be using to solve the problem So the first formula is cos 3a is equal to 4 cos cube a minus 3 cos a and the second one is cos 2a is equal to 2 cos square a minus 1. So this becomes the key idea to solve the problem Let's now proceed on to the solution we find out the LHS which is cos 6x Now we can write cos 6x as cos 3 into 2x Since we want to use this formula Now we apply the formula so that it becomes 4 cos cube 2x because here a is 2x right minus 3 cos 2x Now We apply the formula for cos 2x so that it becomes 2 cos square x minus 1 cube because we have a cube here minus 3 into 2 cos square x minus 1 Now We apply the formula for a minus b whole cube so that it becomes 2 cos square x whole cube minus 3 into 2 cos square x square into 1 plus 3 into 2 cos square x into 1 square minus 1 minus 3 into 2 cos square x is minus 6 cos square x minus 3 into minus 1 is plus 3 this is by the formula for a minus b whole cube is equal to a cube minus 3 a square b plus 3 a b square minus b cube This is the formula which we have used let's now simplify this move further We get 4 into 8 cos to the power 6x because 2 cube is 8 minus 3 into 4 cos to the power 4x plus 6 into cos square x minus 1 minus 6 cos square x plus 3 Now again This becomes equal to 4 into 8 cos to the power 6x minus 12 cos to the power 4x plus 6 cos square x minus 1 minus 6 cos square x plus 3 Now multiplying these with 4 we get 4 8 z is 32 cos to the power 6x 12 4 z is 48 cos to the power 4x 4 into 6 is 24 cos square x minus 4 minus 6 cos square x plus 3 Now this becomes equal to 32 cos to the power 6x minus 48 cos to the power 4x 24 cos square x minus 6 cos square x is 18 cos square x minus 4 plus 3 is minus 1 which is our RHS This is what we had to prove hence the result is proved hence cos 6x is equal to 32 cos 6x minus 48 cos to the power 4x plus 18 cos square x minus 1 Hence proved So this completes the question. Bye for now. Take care. Have a good day