Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.
Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.
We know that,
$\operatorname{cosec}^{2} A=1+\cot ^{2} A$
$\frac{1}{\operatorname{cosec}^{2} A}=\frac{1}{1+\cot ^{2} A}$
$\sin ^{2} A=\frac{1}{1+\cot ^{2} A}$
$\sin A=\pm \frac{1}{\sqrt{1+\cot ^{2} A}}$
$\sqrt{1+\cot ^{2} A}$ will always be positive as we are adding two positive quantities.
Therefore, $\sin A =\frac{1}{\sqrt{1+\cot ^{2} A }}$
We know that, $\tan A =\frac{\sin A }{\cos A }$
However, $\cot A=\frac{\cos A}{\sin A}$
Therefore, $\tan A =\frac{1}{\cot A }$
Also, $\sec ^{2} A=1+\tan ^{2} A$
$=1+\frac{1}{\cot ^{2} A}$
$=\frac{\cot ^{2} A+1}{\cot ^{2} A}$
$\sec A=\frac{\sqrt{\cot ^{2} A+1}}{\cot A}$
Similar Questions
Express the ratios $\cos A ,$ tan $A$ and $\sec A$ in terms of $\sin A .$
Express the ratios $\cos A ,$ tan $A$ and $\sec A$ in terms of $\sin A .$
In $\triangle$ $PQR,$ right-angled at $Q$ (see $Fig.$), $PQ =3 \,cm$ and $PR =6 \,cm$. Determine $\angle QPR$ and $\angle PRQ$.
In $\triangle$ $PQR,$ right-angled at $Q$ (see $Fig.$), $PQ =3 \,cm$ and $PR =6 \,cm$. Determine $\angle QPR$ and $\angle PRQ$.
Evaluate the following:
$\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}$
Evaluate the following:
$\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}$
Evaluate:
$\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$
Evaluate:
$\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$
Show that:
$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ}=1$
$(ii)$ $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}=0$
Show that:
$(i)$ $\tan 48^{\circ} \tan 23^{\circ} \tan 42^{\circ} \tan 67^{\circ}=1$
$(ii)$ $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}=0$