The value of ^2 3A ^2 A - ^2 3A ^2 A is

Step-by-step solution

Using the multiple angle formulas 3A = 3 A - 4 ^3 A and 3A = 4 ^3 A - 3 A . 3A A = 3 - 4 ^2 A 3A A = 4 ^2 A - 3 Substituting these into the given expression: ( 3A A )^2 - ( 3A A )^2 = (3 - 4 ^2 A)^2 - (4 ^2 A - 3)^2 Expanding the squares: = (9 - 24 ^2 A + 16 ^4 A) - (16 ^4 A - 24 ^2 A + 9) = 16( ^4 A - ^4 A) - 24( ^2 A - ^2 A) Using a^2 - b^2 = (a-b)(a+b) and ^2 A + ^2 A = 1 : = 16( ^2 A - ^2 A)( ^2 A + ^2 A) - 24( ^2 A - ^2 A) = 16( ^2 A - ^2 A)(1) - 24( ^2 A - ^2 A) = -8( ^2 A - ^2 A) = 8( ^2 A - ^2 A) Since ^2 A