Prove that:

tan²x + sec²x = 1

tan²x + sec²x = 1

tan = sin / cos

sec = 1/cos

sin^2 x/cos^2 x + 1 / cos ^2 x = 1

since it has a common denominator

(sin ^2 x+1) / cos ^2 x = 1

multiply by cos ^2 x on each side

sin ^2 x + 1 = cos ^2 x

replace 1 = sin^2 + cos ^2

sin ^2 x + sin ^2 x + cos^2 x = cos ^2 x

combine like terms

2 sin ^2 x + cos ^2 = cos ^ 2 x

subtract cos ^2 x from each side

2 sin^2 x = 0

divide by 2 on each side

sin^2 x = 0

take the square root on each side

sin x = 0

take the arcsin on each side

arcsin (sin x) = arcsin (0)

x = 0, pi, 2 * ip

x = n * pi where n is an integer