An identity is an equation that is always true for any value of the variable. To verify that an equation is an identity, we can substitute different values for the variable and see if the equation still holds true.
The given equation is (5sinx+5cosx)^2=25sin2x+25
We can start by expanding the left side of the equation:
(5sinx+5cosx)^2 = (5sinx)^2 + 2(5sinx)(5cosx) + (5cosx)^2
= 25sin^2x + 50sinxcosx + 25cos^2x
= 25(sin^2x + cos^2x) + 50sinxcosx
= 25 + 50sinxcosx
Now we can use the trigonometric identity sin^2x + cos^2x = 1 to simplify the equation further:
25 + 50sinxcosx = 25 + 50(sinxcosx) = 25 + 25sin2x
Now we can see that the left side of the original equation is equal to the right side of the equation, 25sin2x+25.
Therefore, the equation (5sinx+5cosx)^2 = 25sin2x+25 is an identity for all values of x.