1. I have been trying to prove this trigonometric identity:
. . . . .[tex]\dfrac{\tan(\theta)\, +\, \sec(\theta)\, -\, 1}{\tan(\theta)\, -\, \sec(\theta)\, +\, 1}\, =\, \dfrac{1\, +\, \sin(\theta)}{\cos(\theta)}[/tex]

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:
. . . . .[tex]\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)\,}}[/tex]

But, I am stuck in this position. Please help me to solve it further. Please tell, how can I prove the rest?

. . . . .[tex]\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)}[/tex]

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!
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