Use the fundamental identities and appropriate algebraic operations to simplify the expression.
cosx / (1+sinx)
Answer: secx
Can someone please assist as to how I get to the answer?
You must have a typo
cosx/(1+sinx) is NOT equal to secx
try it with any angle.
To simplify the expression cosx / (1+sinx), we can start by using the fundamental identity:
secx = 1 / cosx
From this identity, we can rewrite the denominator (1+sinx) as:
1 + sinx = cos^2x / cosx + sinx = (cos^2x + sinx * cosx) / cosx
Now, we can rewrite the initial expression as:
cosx / (1+sinx) = cosx / (cos^2x + sinx * cosx) / cosx
Simplifying further, we can cancel out the common factors of cosx:
cosx / (cos^2x + sinx * cosx) / cosx = 1 / (cos^2x + sinx * cosx)
Now, let's express sinx * cosx as sinx * cosx = sinx * cosx / cos^2x * cos^2x = tanx / cosx
Substituting this back into the expression:
1 / (cos^2x + sinx * cosx) = 1 / (cos^2x + tanx / cosx)
Using the fundamental identity:
tan^2x + 1 = sec^2x
We can rewrite the expression as:
1 / (cos^2x + tanx / cosx) = 1 / (cos^2x + tan^2x / cos^2x) = 1 / (cos^2x + sec^2x) = 1 / sec^2x = sec^2x
Therefore, the simplified expression is sec^2x (secant squared x).