Question : If $\sec x + \tan x = 5$ and $cosec\; y - \cot y = \frac{1}{3}$, then find the value of $(\sec x + cosec \;y) -(\tan x - \cot y)$.
Option 1: 4.2
Option 2: 3.2
Option 3: 2.2
Option 4: 3.1
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Correct Answer: 3.2
Solution :
$\sec x + \tan x = 5$
$\sec^2x - \tan^2x = 1$
$⇒ (\sec x + \tan x)(\sec x - \tan x) = 1$
$⇒ 5 \times (\sec x - \tan x) = 1$
$⇒ (\sec x - \tan x) = \frac{1}{5}$
Now, $\operatorname{cosec}y - \cot y =\frac{1}{3}$
$\operatorname{cosec}^2y - \cot^2y = 1$
$⇒(\operatorname{cosec}y + \cot y)(\operatorname{cosec}y - \cot y) = 1$
$⇒ \frac{1}{3} \times (\operatorname{cosec} y + \cot y) = 1$
$⇒ \operatorname{cosec} y + \cot y = 3$
Again according to the question,
$(\sec x + \operatorname{cosec} y) - (\tan x - \cot y)$
$= \sec x + \operatorname{cosec} y - \tan x + \cot y$
$= \sec x - \tan x + \operatorname{cosec}y + \cot y$
$= \frac{1}{5} + 3 = 3.2$
∴ The value of $(\sec x + cosec \;y) -(\tan x - \cot y)$ is 3.2.
Hence, the correct answer is 3.2.
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