Question

please help me to solve this question​

Answer 1

Answer:

• To Prove :

\dfrac{ \tan(A) }{1 - \cot(A) } + \dfrac{ \cot(A) }{1 - \tan(A) } = 1 + \sec(A) \csc(A)

1−cot(A)

tan(A)

+

1−tan(A)

cot(A)

=1+sec(A)csc(A)

• Proof :

\leadsto\dfrac{ \tan(A) }{1 - \cot(A) } + \dfrac{ \cot(A) }{1 - \tan(A) }⇝

1−cot(A)

tan(A)

+

1−tan(A)

cot(A)

⠀⠀⠀⠀⋆ tan(A) = sin(A) / cos(A)

⠀⠀⠀⠀⋆ cot(A) = cos(A) / sin(A)

\leadsto\dfrac{ \frac{ \sin(A) }{ \cos(A) } }{1 - \frac{ \cos(A) }{ \sin(A) } } + \dfrac{ \frac{ \cos(A) }{ \sin(A) } }{1 - \frac{ \sin(A) }{ \cos(A) } }⇝

1−

sin(A)

cos(A)

cos(A)

sin(A)

+

1−

cos(A)

sin(A)

sin(A)

cos(A)

\leadsto\dfrac{ \frac{ \sin(A) }{ \cos(A) } }{\frac{ \sin(A) - \cos(A) }{ \sin(A) } } + \dfrac{ \frac{ \cos(A) }{ \sin(A) } }{\frac{ \cos(A) - \sin(A) }{ \cos(A) } }⇝

sin(A)

sin(A)−cos(A)

cos(A)

sin(A)

+

cos(A)

cos(A)−sin(A)

sin(A)

cos(A)

\leadsto\dfrac{ \sin^{2} (A) }{ \cos(A)( \sin(A) - \cos(A)) } + \dfrac{ \cos ^{2} (A) }{ \sin(A)( \cos(A) - \sin(A) )}⇝

cos(A)(sin(A)−cos(A))

sin

2

(A)

+

sin(A)(cos(A)−sin(A))

cos

2

(A)

⠀⠀⠀⠀⋆ Taking Negative Common

\leadsto\dfrac{ \sin^{2} (A) }{ \cos(A)( \sin(A) - \cos(A)) } - \dfrac{ \cos ^{2} (A) }{ \sin(A)( \sin(A) - \cos(A) )}⇝

cos(A)(sin(A)−cos(A))

sin

2

(A)

−

sin(A)(sin(A)−cos(A))

cos

2

(A)

\leadsto \dfrac{1}{( \sin(A) - \cos(A))} \bigg(\dfrac{ \sin^{2} (A) }{ \cos(A) } - \dfrac{ \cos ^{2} (A) }{ \sin(A)} \bigg)⇝

(sin(A)−cos(A))

1

(

cos(A)

sin

2

(A)

−

sin(A)

cos

2

(A)

)

\leadsto \dfrac{1}{( \sin(A) - \cos(A))} \bigg(\dfrac{ \sin^{3} (A) - \cos ^{3} (A) }{ \cos(A)\sin(A) } \bigg)⇝

(sin(A)−cos(A))

1

(

cos(A)sin(A)

sin

3

(A)−cos

3

(A)

)

⠀⠀⠀⠀⋆ (a³ - b³) = (a - b)(a² + b² + ab)

\leadsto \dfrac{1}{ \cancel{( \sin(A) - \cos(A))}} \times \dfrac{ \cancel{(\sin(A) - \cos(A))}(\sin^{2} (A) + \cos ^{2} (A) + \sin(A) \cos(A)) }{ \cos(A)\sin(A) }⇝

(sin(A)−cos(A))

1

×

cos(A)sin(A)

(sin(A)−cos(A))

(sin

2

(A)+cos

2

(A)+sin(A)cos(A))

\leadsto \dfrac{(\sin^{2} (A) + \cos ^{2} (A) + \sin(A) \cos(A)) }{ \cos(A)\sin(A) }⇝

cos(A)sin(A)

(sin

2

(A)+cos

2

(A)+sin(A)cos(A))

⠀⠀⠀⠀⋆ (sin²A + cos²A) = 1

\leadsto \dfrac{(1 + \sin(A) \cos(A)) }{ \cos(A)\sin(A) }⇝

cos(A)sin(A)

(1+sin(A)cos(A))

\leadsto \dfrac{1}{ \cos(A)\sin(A) } + \cancel\dfrac{\sin(A) \cos(A) }{ \cos(A)\sin(A) }⇝

cos(A)sin(A)

1

+

cos(A)sin(A)

sin(A)cos(A)

\leadsto \dfrac{1}{ \cos(A)\sin(A) } + 1⇝

cos(A)sin(A)

1

+1

⠀⠀⠀⠀⋆ 1 / cos(A) = sec(A)

⠀⠀⠀⠀⋆ 1 / sin(A) = cosec(A)

\leadsto \large1 + \sec(A) \csc(A)⇝1+sec(A)csc(A)

⠀

\therefore \boxed{ \rm \dfrac{ \tan(A) }{1 - \cot(A) } + \dfrac{ \cot(A) }{1 - \tan(A) } = 1 + \sec(A) \csc(A) }∴

1−cot(A)

tan(A)

+

1−tan(A)

cot(A)

=1+sec(A)csc(A)

sorry it is too long