If $ \sec 4 A=\operatorname{cosec}\left(A-20^{\circ}\right) $, where $ 4 A $ is an acute angle, find the value of $ A $.
Given:
\( \sec 4 A=\operatorname{cosec}\left(A-20^{\circ}\right) \), where \( 4 A \) is an acute angle.
To do:
We have to find the value of \( A \).
Solution:
We know that,
$\operatorname{cosec} (90^{\circ}- \theta) = sec\ \theta$
Therefore,
$\sec 4 A=\operatorname{cosec}\left(A-20^{\circ}\right)$
$\operatorname{cosec} (90^{\circ}- 4A)=\operatorname{cosec}\left(A-20^{\circ}\right)$
Comparing on both sides, we get,
$90^{\circ}-4A =A-20^{\circ}$
$4A+A=90^{\circ}+20^{\circ}$
$5A=110^{\circ}$
$A=\frac{110^{\circ}}{5}$
$A=22^{\circ}$
The value of \( A \) is $22^{\circ}$.
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