Prove that A (90^ -A ) + A (90^ -A ) = (90^ -A ) e c (90^ -A ) - Vidyadip

Question

$
\text { Prove that } \frac{\sin A}{\sin \left(90^{\circ}-A\right)}+\frac{\cos A}{\cos \left(90^{\circ}-A\right)}=\sec \left(90^{\circ}-A\right) \cos e c\left(90^{\circ}-A\right)
$

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Answer

$
\begin{aligned}
& \text { LHS }=\frac{\sin A}{\sin \left(90^{\circ}-A\right)}+\frac{\cos A}{\cos \left(90^{\circ}-A\right)} \\
& \Rightarrow \frac{\cos \left(90^{\circ}-A\right)}{\sin \left(90^{\circ}-A\right)}+\frac{\sin \left(90^{\circ}-A\right)}{\cos \left(90^{\circ}-A\right)} \\
& \Rightarrow \frac{\cos ^2\left(90^{\circ}-A\right)+\sin ^2\left(90^{\circ}-A\right)}{\sin \left(90^{\circ}-A\right) \cdot \cos \left(90^{\circ}-A\right)}=\frac{1}{\sin \left(90^{\circ}-A\right) \cdot \cos \left(90^{\circ}-A\right)} \\
& \Rightarrow \sec \left(90^{\circ}-A\right) \operatorname{cosec}\left(90^{\circ}-A\right)
\end{aligned}
$

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