Hi there! I NEED SERIOUS HELP, PLEASE!!! i have such a hard time with verifying identities! The question is:
[(sin(theta/2)) / csc(theta/2)] + [(cos (theta/2) / sec(theta/2)] = 1
I have a few ideas on how to solve this, but am mainly not sure how to get rid of (theta/2). I am taking trig online and use professor notes and the textbook, in addition to google searches.
I would normally try to add both fractions if they both had the same denominator, but am unsure how to find a common denominator in this question. Do I even need to be finding a common denominator??
If I am shown how to get rid of (theta/2) to make "2sin theta", or somehow "sin^2 theta", I would then do [(1/csc theta) / (1/sin theta)] + [(1/sec theta) / (1/cos theta)] .... which even then, I am not sure if that will get me anywhere.
OR
Do I cross multiply numerators together and denominators together?? Making it
[(sin(theta/2)cos(theta/2)] / [(csc(theta/2)sec(theta/2)] ?
***I would mainly appreciate a formula to mimic, and any form of help is GREATLY appreciated it. Thank you for your time and help in advance!
Stephani
While this might appear to be an exercise in half-angle or double-angle formulas, it is really just a check to see whether you remember your basic trig definitions:
cscθ = 1/sinθ and secθ = 1/cosθ
You happen to be using θ/2, but the principle holds.
so, since cscθ=1/sinθ, 1/cscθ = sinθ
[(sin(θ/2)) / csc(θ/2)] + [(cos (θ/2) / sec(θ/2)] = 1
[sin(θ/2) * sin(θ/2)] + cos(θ/2) * cos(θ/2)] = 1
sin^2(θ/2) + cos^2(θ/2) = 1
as we all know, this is true.
Hi Stephani! It seems like you're having trouble with verifying identities in trigonometry. Don't worry, I'm here to help!
To solve the given trigonometric equation [(sin(theta/2)) / csc(theta/2)] + [(cos(theta/2) / sec(theta/2)] = 1, we need to simplify the left side of the equation and make it equal to the right side (which is 1).
One approach is to rewrite all trigonometric functions in terms of sine and cosine, which are more commonly used. Let's start by dealing with the fractions:
1. The first fraction is (sin(theta/2)) / csc(theta/2). Remember that csc is the reciprocal of sin, so we can rewrite this fraction as sin(theta/2) / (1/sin(theta/2)). Simplifying further, we get sin(theta/2) * sin(theta/2) = sin^2(theta/2).
2. Similarly, the second fraction is (cos(theta/2)) / sec(theta/2). Recall that sec is the reciprocal of cos, so we can rewrite this fraction as cos(theta/2) / (1/cos(theta/2)). Simplifying further, we get cos(theta/2) * cos(theta/2) = cos^2(theta/2).
Now, our equation becomes sin^2(theta/2) + cos^2(theta/2) = 1, which is a fundamental identity in trigonometry known as the Pythagorean identity. So by using this identity, we can conclude that the left side of the equation is indeed equal to the right side.
To summarize, the first step is to rewrite the given trigonometric functions in terms of sine and cosine, if possible. Then, look for any applicable trigonometric identities that can be used to simplify the equation. In this case, we used the Pythagorean identity sin^2(theta/2) + cos^2(theta/2) = 1.
I hope this explanation helps you understand the process of verifying identities in trigonometry. If you have any further questions, feel free to ask!