Hi! My teacher told my class that the answer to this question is "1-sin x/cosx" but she wants us to figure out why. I tried solving it but I didn't end up with the answer given. Can someone help me? Thank you!
Problem: cos x/1+sinx
A little "trick" needed here
cosx/(1+sinx) , (those brackets are essential)
= cosx/(1+sinx) * (1-sinx)/(1-sinx) ---> I multiplied by a version of 1, thus changing the appearance but not the value of the expression
= cosx(1-sinx)/(1 - sin^2 x)
= cosx(1-sinx)/cos^2 x
= (1-sinx)/cosx ----- again, I need those brackets.
Of course, I'd be happy to help you understand how to solve the problem!
To simplify the expression cos x / (1 + sin x), we can start by multiplying the numerator and denominator by the conjugate of the denominator, which is 1 - sin x. This will help us eliminate the complex fraction.
Here's the step-by-step process:
1. Multiply the numerator and denominator by the conjugate of the denominator, (1 - sin x):
cos x * (1 - sin x) / (1 + sin x) * (1 - sin x)
2. Apply the distributive property by multiplying cos x with both terms inside the parentheses:
cos x - cos x * sin x / (1 + sin x) * (1 - sin x)
3. Simplify by multiplying the remaining terms:
cos x - cos x * sin x / (1 - sin^2 x)
4. Recall the Pythagorean identity: sin^2 x + cos^2 x = 1. Rearrange and solve for cos^2 x:
cos^2 x = 1 - sin^2 x
5. Substitute this into the expression:
cos x - cos x * sin x / cos^2 x
6. Simplify by canceling out the common factor of cos x in the numerator and denominator:
(cos x / cos x) - sin x / cos x
7. Finally, simplify to get the desired answer:
1 - sin x / cos x
So, the simplified expression of cos x / (1 + sin x) is indeed 1 - sin x / cos x, as your teacher mentioned.
I hope this explanation clarifies how to arrive at the given answer. If you have any further questions, feel free to ask!