Prove that 2cosx/cos2x+1=secx based on the grade 12 university curriculum of Ontario Canada

Answer 1

We need to show that

2cos(x)/(cos(2x) + 1) = sec(x).

We can start by rewriting sec(x) in terms of cos(x):

sec(x) = 1/cos(x).

Now, let's rewrite cos(2x) using the double angle formula:

cos(2x) = 2cos^2(x) - 1.

Substitute this back into the expression:

2cos(x)/(2cos^2(x) - 1 + 1)
= 2cos(x)/(2cos^2(x))
= 1/cos(x)
= sec(x).

Therefore, the expression 2cos(x)/(cos(2x) + 1) is equivalent to sec(x), which proves the statement.