To simplify cos2 x (sec2x + sec2 x tan2x), we can use the trigonometric identity: 1 + tan2 x = sec2 x.
First, we can simplify sec2 x + sec2 x tan2 x:
sec2 x + sec2 x tan2 x = sec2 x (1 + tan2 x)
Next, we can substitute sec2 x (1 + tan2 x) into the original equation:
cos2 x (sec2 x + sec2 x tan2 x) = cos2 x (sec2 x (1 + tan2 x))
Using the identity 1 + tan2 x = sec2 x, we can simplify:
cos2 x (sec2 x (1 + tan2 x)) = cos2 x (sec2 x)(sec2 x)
Finally, using the identity sec2 x = 1/cos2 x, we can simplify:
cos2 x (sec2 x)(sec2 x) = cos2 x (1/cos2 x)(1/cos2 x) = 1/cos2 x = sec2 x
Therefore, the answer is 1+tan2xsec2x.