Question

if cos(x)=3/5, find sec(pi/2-x)
I don't know how to get to the answer
And because I also want to know how to get to the answer on a simpler version of the same question,
if sec(x)=3/2, find tan(x)

Answer 1

cos(π/2 - x) = sin(x)
The co- in cosine means sine of the complement
So, sec(π/2 - x) = csc(x)

One of the most useful of the many trig identities is
sin^2x + cos^2x = 1
Divide that by cos^2x and you get
tan^2x + 1 = sec^2x
So, if secx = 3/2, tan^2x = 9/4 - 1 = 5/4
so tanx = √5/2

An easier way to do conversions like this just to draw the right triangle involved, using the Pythagorean Theorem.
Hypotenuse = 3
Adjacent leg = 2
so opposite leg = √(3^2-2^2) = √5
tanx = opposite/adjacent = √5/2