Question

Prove that tan theta /1-cot theta + cot theta/1- tan theta =1+ sec theta × cos theta

Answer 1

heya..!!!!


tan/1-cot∅ + cot∅/1-tan∅

=> sin∅/cos∅/(sin∅-cos∅)/sin∅ + cos∅/sin∅/(cos∅-sin∅)/cos∅

=> sin²∅/cos∅(sin∅-cos∅) + cos²∅/sin∅(cos∅-sin∅)

=> (sin³∅ - cos³∅) / sin∅cos∅(sin∅-cos∅)

=> [ (sin∅-cos∅)(sin²∅+cos²∅+sin∅cos∅) ] /sin∅cos∅(sin∅-cos∅)

=> (1+sin∅cos∅) / sin∅cos∅

=> 1/sin∅cos∅ + sin∅cos∅/sin∅cos∅

=> sec∅cosec∅ + 1

Answer 2

Step-by-step explanation:

$\huge{\pink{\boxed{\green{\underline{\red{\sf{SOLUTION-}}}}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: { \orange {TO \: PROVE}} \\ { \pink{ \boxed{ \red{\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = 1 + \sec(a) \times \cosec(a) }}}}$

☆ USING TRIGONOMETRIC IDENTITIES:

$( \theta=a)\\\to\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) }{1 - \tan(a) } = 1 + \sec(a) \times \cosec(a) \\ \: \: \: \: \: \: \: \: \: LHS \\ \to\frac{ \tan(a) }{1 - \cot(a) } + \frac{ \cot(a) ) }{1 - \tan(a) } \\ \to \frac{ \frac{ \sin(a) }{ \cos(a) } }{ 1 - \frac{ \cos(a) }{\sin(a) } } + \frac{ \frac{ \cos(a) }{ \sin(a) } }{1 - \frac{ \sin(a) }{ \cos(a) } } \\ \to \frac{ \frac{ \sin(a) }{ \cos(a) } }{ \frac{ \sin(a) - \cos(a) }{ \sin(a) } } + \frac{ \frac{ \cos(a) }{ \sin(a) } }{ \frac{ \cos(a) - \sin(a) }{ \cos(a) } } \\ \to \frac{ \sin ^{2} (a) }{ \cos(a) ( \sin(a) - \cos(a) ) } - \frac{ \cos ^{2} (a) }{ \sin(a)( \sin(a) - \cos(a) ) } \\ \to \frac{1} {( \sin(a) - \cos(a)) } ( \frac{ \sin ^{2} (a) }{ \cos(a) } - \frac{ \cos^{2} (a) }{ \sin(a) } ) \\ \to (\frac{1}{ \sin(a) - \cos(a) } )( \frac{ \sin ^{3} (a) - \cos ^{3} (a) }{ \sin(a) \times \cos(a) } ) \\ \to (\frac{1}{ \sin(a) - \cos(a) } )( \frac{( \sin(a) - \cos(a))( \sin ^{2} (a) + \cos ^{2} (a) + \sin(a) \times \cos(a) }{ \sin(a) \times \cos(a) } ) \\ \to \frac{1 + \sin(a) \times \cos(a) }{( \sin(a) \times \cos(a) )} \\ \to 1 + \sec(a) \times \cosec(a) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: LHS = RHS$

$\huge{\pink{\boxed{\green{\underline{\sf{VERIFIED}}}}}}$

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