Question
Without using trigonometric identity, show that:
$
\cos ^2 25^{\circ}+\cos ^2 65^{\circ}=1
$
$
\cos ^2 25^{\circ}+\cos ^2 65^{\circ}=1
$
✓
Answer
$
\begin{aligned}
& \text { Consider } \cos ^2 25^{\circ}+\cos ^2 65^{\circ} \\
& \Rightarrow \cos ^2\left(90^{\circ}-65^{\circ}\right)+\cos ^2 65^{\circ} \\
& \Rightarrow \sin ^2 65^{\circ}+\cos ^2 65^{\circ}=1
\end{aligned}
$
\begin{aligned}
& \text { Consider } \cos ^2 25^{\circ}+\cos ^2 65^{\circ} \\
& \Rightarrow \cos ^2\left(90^{\circ}-65^{\circ}\right)+\cos ^2 65^{\circ} \\
& \Rightarrow \sin ^2 65^{\circ}+\cos ^2 65^{\circ}=1
\end{aligned}
$
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\tan 10^{\circ} \tan 20^{\circ} \tan 30^{\circ} \tan 70^{\circ} \tan 80^{\circ}=\frac{1}{\sqrt{3}}
$
→$
\tan 10^{\circ} \tan 20^{\circ} \tan 30^{\circ} \tan 70^{\circ} \tan 80^{\circ}=\frac{1}{\sqrt{3}}
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