Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$
$LHS =\frac{\cot A -\cos A }{\cot A +\cos A }=\frac{\frac{\cos A }{\sin A }-\cos A }{\frac{\cos A }{\sin A }+\cos A }$
$=\frac{\cos A\left(\frac{1}{\sin A}-1\right)}{\cos A\left(\frac{1}{\sin A}+1\right)}=\frac{\left(\frac{1}{\sin A}-1\right)}{\left(\frac{1}{\sin A}+1\right)}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}=R H S$
Similar Questions
State whether the following are true or false. Justify your answer.
$(i)$ The value of tan $A$ is always less than $1 .$
$(ii)$ $\sec A=\frac{12}{5}$ for some value of angle $A$.
State whether the following are true or false. Justify your answer.
$(i)$ The value of tan $A$ is always less than $1 .$
$(ii)$ $\sec A=\frac{12}{5}$ for some value of angle $A$.
Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of
$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$
$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$
Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of
$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$
$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$
If $\sin A =\frac{3}{4},$ calculate $\cos A$ and $\tan A$.
If $\sin A =\frac{3}{4},$ calculate $\cos A$ and $\tan A$.
Express $\sin 67^{\circ}+\cos 75^{\circ}$ in terms of trigonometric ratios of angles between $0^{\circ}$ and $45^{\circ}$
Express $\sin 67^{\circ}+\cos 75^{\circ}$ in terms of trigonometric ratios of angles between $0^{\circ}$ and $45^{\circ}$
If $\sin ( A - B )=\frac{1}{2}, \cos ( A + B )=\frac{1}{2}, 0^{\circ} < A + B \leq 90^{\circ}, A > B ,$ find $A$ and $B$
If $\sin ( A - B )=\frac{1}{2}, \cos ( A + B )=\frac{1}{2}, 0^{\circ} < A + B \leq 90^{\circ}, A > B ,$ find $A$ and $B$