if-tan-x-sec-x-7-8-find-cot-x-cosec-x-

Question Number 81963 by jagoll last updated on 17/Feb/20

i f \tan (x) + \sec (x) = \frac{7}{8}

f i n d \cot (x) + cosec (x) =

Commented by john santu last updated on 17/Feb/20

l e t \cot (x) + cosec (x) = t

(i) {\tan (x) + \sec (x)} \times {\cot (x) + cosec (x)} = \frac{7 t}{8}

1 + \frac{1}{\cos (x)} + \frac{1}{\sin (x)} + \frac{1}{\sin (x) \cos (x)} = \frac{7 t}{8}

(i i) - \frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} = - \frac{7}{8}

(i i i) - \frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} = - t

(i) + (i i) + (i i i)

\Rightarrow 1 = - \frac{t}{8} - \frac{7}{8}

\Rightarrow - 8 = t + 7 \Rightarrow t = - 15

∴ \cot (x) + cosec (x) = - 15

Commented by jagoll last updated on 17/Feb/20

t h a n k y o u m i s t e r

Answered by Kunal12588 last updated on 17/Feb/20

l e t t a n x = t

s e c x = \sqrt{1 + t^{2}}

c o t x = \frac{1}{t}

c s c x = \frac{\sqrt{1 + t^{2}}}{t}

t + \sqrt{1 + t^{2}} = \frac{7}{8}

\Rightarrow 1 + t^{2} = \frac{49}{64} + t^{2} - \frac{7}{4} t

\Rightarrow \frac{7}{4} t = \frac{49}{64} - 1 = \frac{- 15}{64}

\Rightarrow t = \frac{- 15}{112}

\sqrt{1 + t^{2}} = \frac{7}{8} - t = \frac{7}{8} + \frac{15}{112} = \frac{98 + 15}{112} = \frac{113}{112}

\frac{1}{t} + \frac{\sqrt{1 + t^{2}}}{t} = \frac{1 + \sqrt{1 + t^{2}}}{t} = \frac{1 + \frac{113}{112}}{\frac{- 15}{112}} = - \frac{225}{15} = - 15

c o t x + c s c x = - 15

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