# Thread: prove (tan@+sec@-1)/(tan@-sec@+1)=(1+sin@)/(cos@) by method different from book's

1. ## prove (tan@+sec@-1)/(tan@-sec@+1)=(1+sin@)/(cos@) by method different from book's

1. I have been trying to prove this trigonometric identity:

. . . . .$\dfrac{\tan(\theta)\, +\, \sec(\theta)\, -\, 1}{\tan(\theta)\, -\, \sec(\theta)\, +\, 1}\, =\, \dfrac{1\, +\, \sin(\theta)}{\cos(\theta)}$

My textbook proves it using two different methods. See them here.

Now, I made an attempt to solve this question in a different way. I have somehow reached this step:

. . . . .$\mbox{L.H.S. }\, =\, \dfrac{\tan(\theta)\, +\, \sec(\theta)\, -\, 1}{\tan(\theta)\, -\, \sec(\theta)\, +\, 1}\, =\, \dfrac{\sqrt{\sin(\theta)\, +\, 1\,}}{\sqrt{1\, -\, \cos(\theta)\,}}\, \cdot\, \dfrac{\sqrt{1\, +\, \sin(\theta)\,}\, -\, \sqrt{1\, -\, \sin(\theta)\,}}{\sqrt{1\, +\, \cos(\theta)\,}\, -\, \sqrt{1\, -\, \cos(\theta)\,}}$

. . . . .$\dfrac{\sqrt{\sin(\theta)\, +\, 1\,}}{\sqrt{1\, -\, \cos(\theta)\,}}\, \cdot\, \dfrac{\sqrt{1\, +\, \sin(\theta)\,}\, -\, \sqrt{1\, -\, \sin(\theta)\,}}{\sqrt{1\, +\, \cos(\theta)\,}\, -\, \sqrt{1\, -\, \cos(\theta)\,}}\, =\, \dfrac{1\, +\, \sin(\theta)}{\cos(\theta)}$

I also want to know one more thing,

2. By how many different methods can a single trigonometric identity be proved? (one, two, or infinitely many)

I will be thankful for help!

2. If A/B = C/D then A*D = B*C. Prove the second equality instead. Seems like this approach was not used.

Regarding your other question, there can be many proofs. It's difficult to say how many. How would you formally classify them? Is proof #50 a new approach or just a slight tweak of proof #35? Is this question from the book?

3. There are many proofs for the Pythagorean Theorem. Here are 118. https://www.cut-the-knot.org/pythagoras/

4. Originally Posted by navneet9431
1. I have been trying to prove this trigonometric identity:

. . . . .$\dfrac{\tan(\theta)\, +\, \sec(\theta)\, -\, 1}{\tan(\theta)\, -\, \sec(\theta)\, +\, 1}\, =\, \dfrac{1\, +\, \sin(\theta)}{\cos(\theta)}$

My textbook proves it using two different methods. See them here.

Now, I made an attempt to solve this question in a different way. I have somehow reached this step:

. . . . .$\mbox{L.H.S. }\, =\, \dfrac{\tan(\theta)\, +\, \sec(\theta)\, -\, 1}{\tan(\theta)\, -\, \sec(\theta)\, +\, 1}\, =\, \dfrac{\sqrt{\sin(\theta)\, +\, 1\,}}{\sqrt{1\, -\, \cos(\theta)\,}}\, \cdot\, \dfrac{\sqrt{1\, +\, \sin(\theta)\,}\, -\, \sqrt{1\, -\, \sin(\theta)\,}}{\sqrt{1\, +\, \cos(\theta)\,}\, -\, \sqrt{1\, -\, \cos(\theta)\,}}$

. . . . .$\dfrac{\sqrt{\sin(\theta)\, +\, 1\,}}{\sqrt{1\, -\, \cos(\theta)\,}}\, \cdot\, \dfrac{\sqrt{1\, +\, \sin(\theta)\,}\, -\, \sqrt{1\, -\, \sin(\theta)\,}}{\sqrt{1\, +\, \cos(\theta)\,}\, -\, \sqrt{1\, -\, \cos(\theta)\,}}\, =\, \dfrac{1\, +\, \sin(\theta)}{\cos(\theta)}$

I also want to know one more thing,

2. By how many different methods can a single trigonometric identity be proved? (one, two, or infinitely many)

I will be thankful for help!
I am Khan-fused with all those √ signs in your work.

If I were to do this problem, I would start as follows:

$\displaystyle{\dfrac{tan\theta + sec\theta - 1}{tan\theta - sec\theta + 1}}$

=$\displaystyle{\dfrac{sin\theta - cos\theta + 1}{sin\theta + cos\theta - 1}}$

=$\displaystyle{\dfrac{sin\theta - cos\theta + 1}{sin\theta + cos\theta - 1} * \dfrac{sin\theta + (1 - cos\theta)}{sin\theta + (1 - cos\theta)}}$

=$\displaystyle{\dfrac{(sin\theta - cos\theta + 1)^2}{sin^2\theta - (1- cos\theta)^2}}$

and continue.....