Question

The value of $\left(cosec{36}^{\circ }\right)\left(sec{54}^{\circ }\right)-\left(tan{54}^{\circ }\right)\left(cot{36}^{\circ }\right)is:$

• A. 0.0
• B. - 1
• C. 1
• D. 2

Answer 1

The correct option is C 1

Since $cosec({90}^{\circ }-x)=secxandcot({90}^{\circ }-x)=tanx$,
we have:
$cosec\left({36}^{\circ }\right)=cosec({90}^{\circ }-{54}^{\circ })=sec\left({54}^{\circ }\right)---\left(i\right)$
$cot\left({36}^{\circ }\right)=cot({90}^{\circ }-{54}^{\circ })=tan\left({54}^{\circ }\right)---\left(ii\right)$

Substituting (i) and (ii) in the given problem, we get:
sec $\left({54}^{\circ }\right)sec\left({54}^{\circ }\right)-tan\left({54}^{\circ }\right)tan\left({54}^{\circ }\right)$
$=se{c}^2\left({54}^{\circ }\right)-ta{n}^2\left({54}^{\circ }\right)$
$=1[Sincese{c}^{2}x-ta{n}^{2}x=1]$