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Question Number 11844 by minakshidahaval0202@gmail.co. last updated on 02/Apr/17

\cot α + cosec α = k t h e n f i n d cosec α - c o t α a n d a l s o f i n d c o t α

Answered by sma3l2996 last updated on 02/Apr/17

w e h a v e

(i) : c o t α + c o s e c α = \frac{c o s α}{s i n α} + \frac{1}{s i n α} = k

= \frac{c o s α + 1}{s i n α} = \frac{(c o s α + 1) (c o s α - 1)}{s i n α (c o s α - 1)} = \frac{- {s i n}^{2} α}{s i n α (c o s α - 1)}

= \frac{1}{\frac{1}{s i n α} - \frac{c o s α}{s i n α}} = \frac{1}{c o s e c α - c o t α} = k

(i i) : c o s e c α - c o t α = \frac{1}{k}

l e t d o

(i) - (i i) : 2 c o t α = k - \frac{1}{k}

c o t α = \frac{k^{2} - 1}{k}

Commented by sma3l2996 last updated on 02/Apr/17

I m e a n c o t α = \frac{k^{2} - 1}{2 k}

Answered by ajfour last updated on 02/Apr/17

cosec^{2} α - \cot^{2} α = 1

(cosec α - \cot α) (cosec α + \cot α) = 1

(cosec α - \cot α) (k) = 1

cosec α - \cot α = \frac{1}{k} \dots (i i)

a n d a s cosec α + \cot α = k \dots (i)

(i) - (i i) g i v e s

2 \cot α = k - \frac{1}{k}

\cot α = \frac{k^{2} - 1}{2 k} .

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