Question 6:
Since we are doing proofs, we can only work with one side at a time.
I'll work with the left side. Also, I'll be using x instead of theta, since its easier to type.
(sinē x) / (1 - cos x)
Pythagorean Identity: sinē x + cosē x = 1
Rearrange it so it looks like: sinē x = 1 - cosē x
Substitute it into the proof.
(1 - cosē x) / (1 - cos x)
(1 - cosē x) is difference of squares so factor it out.
(1 - cosē x) = (1 - cos x)(1 + cos x)
[(1 - cos x)(1 + cos x)] / (1 - cos x)
Cancel the like terms and you'll have the following.
1 + cos x
Reciprocal Identity: sec x = (1 / cos x)
Rearrange it so it looks like: cos x = (1 / sec x)
Substitute it into the proof.
1 + (1 / sec x)
Combine both terms into one fraction by multiplying 1 by LCD.
(sec x + 1) / (sec x)
Therefore, sinē x / (1 - cos x) = (sec x + 1) / sec x
|