If x = 4cosA + 5sinA and y = 4 sinA - 5 cosA,then the value ofx2 +y2 is

The given equations are x = 4cosA + 5sinA and y = 4sinA - 5cosA.

We need to find the value of x² + y².

Let's start by squaring both equations:

x² = (4cosA + 5sinA)² = 16cos²A + 40cosAsinA + 25sin²A

y² = (4sinA - 5cosA)² = 16sin²A - 40cosAsinA + 25cos²A

Now, let's add these two equations:

x² + y² = 16cos²A + 40cosAsinA + 25sin²A + 16sin²A - 40cosAsinA + 25cos²A

Notice that the term 40cosAsinA cancels out:

x² + y² = 16cos²A + 25sin²A + 16sin²A + 25cos²A

We can group the like terms together:

x² + y² = (16cos²A + 25cos²A) + (25sin²A + 16sin²A)

x² + y² = 41cos²A + 41sin²A

We know that sin²A + cos²A = 1, so we can factor out the 41:

x² + y² = 41(sin²A + cos²A)

x² + y² = 41*1

So, x² + y² = 41.