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prove that :
[tex]\small \frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = \sec(a) \csc(a) + 1 \\ [/tex]

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Sagot :

DᴀʀᴋPᴀʀᴀᴅᴏx

[tex]\qquad \qquad\huge \underline{\boxed{\sf Answer}}[/tex]

The required result is ~

[tex] \sf\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = \sec(a) \csc(a) + 1 \\[/tex]

Refer to the attachment for solution !

View image DᴀʀᴋPᴀʀᴀᴅᴏx
MrRoyal

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

Read more about trigonometry identity at:

https://brainly.com/question/7331447

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