Given that sin(x)^2 + cos(x)^2 = 1, show that sec(x)^2 - tan(x)^2 = 1 (2 marks). Hence solve for x: tan(x)^2 + cos(x) = 1, x ≠ (2n + 1)π and -2π < x =< 2π(3 marks)
sin(x)2 + cos(x)2 = 1
Dividing by cos(x)2 gives:
tan(x)2 + 1 = sec(x)2
Which rearranges as:
sec(x)2 - tan(x)2 = 1 as required.
tan(x)2 + cos(x)2 = 1
sec(x)2 - 1 + cos(x)2 = 1
sec(x)2 + cos(x)2 = 2
1 + cos(x)4 = 2cos(x)2
(cos(x)2 -1)2 = 0
cos(x)2 = 1
cos(x) = 1
x = 0, 2π
