The trigonometry identity [tex]\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = \sec(a) \csc(a) + 1[/tex] is true
The trigonometry identity is given as:
[tex]\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = \sec(a) \csc(a) + 1[/tex]
Express cot(a) as 1/tan(a)
[tex]\frac{ \tan(a) }{1 - \frac{1}{\tan(a)} } + \frac{ \frac{1}{\tan(a)} }{1 - \tan(a) } = \sec(a) \csc(a) + 1[/tex]
Take LCM
[tex]\frac{ \tan(a) }{ \frac{\tan(a) - 1}{\tan(a)} } + \frac{ \frac{1}{\tan(a)} }{1 - \tan(a) } = \sec(a) \csc(a) + 1[/tex]
Simplify the expression
[tex]\frac{ \tan^2(a) }{\tan(a) - 1 } + \frac{ \frac{1}{\tan(a)} }{1 - \tan(a) } = \sec(a) \csc(a) + 1[/tex]
Further, simplify
[tex]\frac{ \tan^2(a) }{\tan(a) - 1 } + \frac{1}{\tan(a)(1 - \tan(a)} = \sec(a) \csc(a) + 1[/tex]
Rewrite as:
[tex]\frac{ \tan^2(a) }{\tan(a) - 1 } - \frac{1}{\tan(a)(\tan(a) -1) } = \sec(a) \csc(a) + 1[/tex]
Take LCM
[tex]\frac{\tan^3(a) - 1}{\tan(a)(\tan(a) -1) } = \sec(a) \csc(a) + 1[/tex]
Expand the numerator
[tex]\frac{(\tan(a) - 1)(\tan^2(a) + \tan(a) + 1)}{\tan(a)(\tan(a) -1) } = \sec(a) \csc(a) + 1[/tex]
Cancel out the common factors
[tex]\frac{\tan^2(a) + \tan(a) + 1}{\tan(a) } = \sec(a) \csc(a) + 1[/tex]
Simplify
[tex]\tan(a) + 1 + \cot(a) = \sec(a) \csc(a) + 1[/tex]
Rewrite as:
[tex]\tan(a) + \cot(a)+ 1 = \sec(a) \csc(a) + 1[/tex]
Rewrite as:
[tex]\frac{\sin(a)}{\cos(a)} + \frac{\cos(a)}{\sin(a)}+ 1 = \sec(a) \csc(a) + 1[/tex]
Take LCM
[tex]\frac{\sin^2(a)+\cos^2(a)}{\cos(a)\sin(a)}+ 1 = \sec(a) \csc(a) + 1[/tex]
Express the numerator as 1
[tex]\frac{1}{\cos(a)\sin(a)}+ 1 = \sec(a) \csc(a) + 1[/tex]
Simplify
[tex]\sec(a) \csc(a) + 1 = \sec(a) \csc(a) + 1[/tex]
Hence, the trigonometry identity [tex]\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = \sec(a) \csc(a) + 1[/tex] has been proved
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