Question

Find the value of $\frac{se{c}^{2}x-cose{c}^{2}x}{ta{n}^{2}x-co{t}^{2}x}$. (x $ϵ$ (0,$\frac{\Pi }{2}$),x $\ne$ $\frac{\Pi }{4}$)

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Answer 1

In the numerator we have secx and cosecx and in the denominator we have tanx and cotx. So we will try to express numerator in terms of denominator. Using the identities,

$Se{c}^{2}x$ = 1 + $ta{n}^{2}x$ ________ (1)

$Cose{c}^{2}x$ = 1 + $co{t}^{2}x$ ________ (2)

(1)-(2) = numerator = $se{c}^{2}x$ - $cose{c}^{2}x$ = $ta{n}^{2}x$ - $co{t}^{2}x$

$\frac{se{c}^{2}x-cose{c}^{2}x}{ta{n}^{2}x-co{t}^{2}x}$ = $\frac{ta{n}^{2}x-co{t}^{2}x}{ta{n}^{2}x-co{t}^{2}x}$ = 1

(Here, since x $ϵ$ $(0,\frac{\Pi }{2})$ and x $\ne$ $\frac{\Pi }{4}$ $ta{n}^{2}x$ - $co{t}^{2}x$ won't be zero.)

Key steps: (1) Identifying the relation between numerator and denominator

(2) Basic identities $se{c}^{2}x$ = 1 + $ta{n}^{2}x$ and $cose{c}^{2}x$ = 1 + $co{t}^{2}x$