if sinx=cos^2x then write the value of cos^2x(1+cos^2x)
To find the value of cos^2x(1+cos^2x) given that sinx = cos^2x, we can first express cos^2x in terms of sinx.
Using the Pythagorean identity sin^2x + cos^2x = 1, we know that cos^2x = 1 - sin^2x.
Therefore, cos^2x = 1 - sin^2x.
Now, substituting this expression into cos^2x(1+cos^2x), we have:
cos^2x(1+cos^2x) = (1 - sin^2x)(1 + (1 - sin^2x)).
Simplifying this expression:
(1 - sin^2x)(1 + (1 - sin^2x)) = (1 - sin^2x)(2 - sin^2x)
Expanding further:
(1 - sin^2x)(2 - sin^2x) = 2 - sin^2x - 2sin^2x + sin^4x.
Combining like terms:
2 - sin^2x - 2sin^2x + sin^4x = 2 - 3sin^2x + sin^4x.
Now, since we know that sinx = cos^2x, we can replace sin^2x with cos^4x:
2 - 3sin^2x + sin^4x = 2 - 3cos^4x + cos^8x.
Thus, the value of cos^2x(1+cos^2x) when sinx = cos^2x is 2 - 3cos^4x + cos^8x.