Question

$\Large{\underline{\underline{\bf{Question:-}}}}$

$\frac{ \cos A - \sin A + 1 }{\cos A + \sin A - 1 }$ = cosec A + cot A, Using the identity cosec² A = 1 + cot² A.​

Answer 1

$\implies\sf \dfrac{ \cos(a) - \sin(a) + 1}{ \cos(a) + \sin(a) - 1 } \\ \\ \implies\dfrac{ \dfrac{ \cos(a) - \sin(a) + 1 }{ \sin(a) } }{ \dfrac{ \cos(a) + \sin(a) - 1}{ \sin(a) } } \\ \\ \implies\dfrac{ \dfrac{ \cos(a) }{ \sin(a) } - \dfrac{ \sin(a) }{ \sin(a) } + \dfrac{1}{ \sin(a) } }{ \ \dfrac{ \cos(a) }{ \sin(a) } + \dfrac{ \sin(a) }{ \sin(a) } - \dfrac{1}{ \sin(a) } } \\ \\ \implies \frac{ \cot(a) - 1 + \csc(a) }{ \cot(a) + 1 - \csc(a) } \\ \\ \implies\frac{ \cot(a) + \csc(a) - ( { \csc(a) }^{2} - { \cot(a) }^{2}) }{ \cot(a) + 1 - \csc(a) } \\ \\ \implies\frac{ \cot(a) + \csc(a) - ( \csc(a) + \cot(a))( \csc(a) - \cot(a) }{ \cot(a) + 1 - \csc(a) } \\ \\ \implies\frac{ \cot(a) + \csc(a)(1 - \csc(a) + \cot(a) }{1 - \csc(a) + \cot(a) } \\ \\ \implies \csc(a) + \cot(a)$

Additional information:-

• Sin²A + Cos²A = 1
• Cosec²A-Cot²A = 1
• Sec²A-Tan²A = 1

Note:- I've used (a) instead of (A)

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hope it's beneficial ⭐

Answer 2

⟹ cos(a)+sin(a)−1

cos(a)−sin(a)+1

⟹ sin(a)cos(a)+sin(a)−1

sin(a)cos(a)−sin(a)+1

⟹ sin(a)cos(a) + sin(a)sin(a) − sin(a)1

sin(a)cos(a)− sin(a)sin(a)+ sin(a)1

⟹ cot(a)+1−csc(a)

cot(a)−1+csc(a)

⟹ cot(a)+1−csc(a)

cot(a)+csc(a)−(csc(a) 2−cot(a) 2 )

⟹ cot(a)+1−csc(a)

cot(a)+csc(a)−(csc(a)+cot(a))(csc(a)−cot(a)

⟹ 1−csc(a)+cot(a)

cot(a)+csc(a)(1−csc(a)+cot(a)

⟹csc(a)+cot(a)

Additional information:-

Sin²A + Cos²A = 1

Cosec²A-Cot²A = 1

Sec²A-Tan²A = 1

Note:- I've used (a) instead of (A)

___________________________

hope it's helps⭐