Which of the following options is equal to c o t θ + c o s e c θ − 1 c o t θ − c o s e c θ + 1 Options: A . &nbsp 1 B . &nbsp s i n θ 1 − c o s θ C . &nbsp 1 + c o s θ s i n θ D . &nbsp cot θ + c o s e c θ

Answer: Option A : B, C, and D = cot θ + c o s e c θ − 1 cot θ − c o s e c θ + 1 = c o t θ + c o s e c θ − ( c o s e c 2 θ − c o t 2 θ ) c o t θ − c o s e c θ + 1 = ( c o t θ + c o s e c θ ) − ( c o s e c θ − c o t θ ) ( c o s e c θ + c o t θ ) c o t θ − c o s e c θ + 1 = ( c o t θ + c o s e c θ ) ( 1 − c o s e c θ + c o t θ ) ( c o t θ − c o s e c θ + 1 ) = c o t θ + c o s e c θ = c o s θ s i n θ + 1 s i n θ = 1 + c o s θ s i n θ = ( 1 + c o s θ ) ( 1 − c o s θ ) s i n θ ( 1 − c o s θ ) ....[Multiplying numerator and denominator by 1 − cos θ ] = 1 − c o s 2 θ s i n θ ( 1 − c o s θ ) = s i n 2 θ s i n θ ( 1 − c o s θ ) = s i n θ 1 − c o s θ

Which of the following options is equal to cot θ+cosec θ

Question

Which of the following options is equal to $\frac{c o t θ + c o s e c θ - 1}{c o t θ - c o s e c θ + 1}$

Options:

A .

1

B .

\frac{s i n θ}{1 - c o s θ}

C .

\frac{1 + c o s θ}{s i n θ}

D .

cot θ + c o s e c θ

Answer: Option A
:
B, C, and D

$= \frac{cot θ + c o s e c θ - 1}{cot θ - c o s e c θ + 1}$
$= \frac{c o t θ + c o s e c θ - (c o s e c^{2} θ - c o t^{2} θ)}{c o t θ - c o s e c θ + 1}$
$= \frac{(c o t θ + c o s e c θ) - (c o s e c θ - c o t θ) (c o s e c θ + c o t θ)}{c o t θ - c o s e c θ + 1}$
$= \frac{(c o t θ + c o s e c θ) (1 - c o s e c θ + c o t θ)}{(c o t θ - c o s e c θ + 1)}$
$= c o t θ + c o s e c θ$
$= \frac{c o s θ}{s i n θ} + \frac{1}{s i n θ}$
$= \frac{1 + c o s θ}{s i n θ}$
$= \frac{(1 + c o s θ) (1 - c o s θ)}{s i n θ (1 - c o s θ)}$ ....[Multiplying numerator and denominator by $1 - cos θ$ ]
$= \frac{1 - c o s^{2} θ}{s i n θ (1 - c o s θ)}$
$= \frac{s i n^{2} θ}{s i n θ (1 - c o s θ)}$
$= \frac{s i n θ}{1 - c o s θ}$