Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
$(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$
$L.H.S.=(\operatorname{cosec} A-\sin A)(\sec A-\cos A)$
$=\left(\frac{1}{\sin A}-\sin A\right)\left(\frac{1}{\cos A}-\cos A\right)$
$=\left(\frac{1-\sin ^{2} A}{\sin A}\right)\left(\frac{1-\cos ^{2} A}{\cos A}\right)$
$=\frac{\left(\cos ^{2} A\right)\left(\sin ^{2} A\right)}{\sin A \cos A}$
$=\sin A \cos A$
$R.H.S=\frac{1}{\tan A+\cot A}$
$=\frac{1}{\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}}=\frac{1}{\sin ^{2} A+\cos ^{2} A}{\sin A \cos A}$
$=\frac{\sin A \cos A}{\sin ^{2} A+\cos ^{2} A}=\sin A \cos A$
Hence,$L . H . S=R . H . S$
Similar Questions
If $\tan A =\cot B ,$ prove that $A + B =90^{\circ}$
If $\tan A =\cot B ,$ prove that $A + B =90^{\circ}$
In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:
$(i)$ $\sin A, \cos A$
$(ii)$ $\sin C, \cos C$
In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:
$(i)$ $\sin A, \cos A$
$(ii)$ $\sin C, \cos C$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\sqrt{\frac{1+\sin A }{1-\sin A }}=\sec A +\tan A$
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$\sqrt{\frac{1+\sin A }{1-\sin A }}=\sec A +\tan A$
Evaluate:
$\frac{\tan 26^{\circ}}{\cot 64^{\circ}}$
Evaluate:
$\frac{\tan 26^{\circ}}{\cot 64^{\circ}}$
State whether the following are true or false. Justify your answer.
$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$
$(ii)$ cot $A$ is the product of cot and $A$.
$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.
State whether the following are true or false. Justify your answer.
$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$
$(ii)$ cot $A$ is the product of cot and $A$.
$(iii)$ $\sin \theta=\frac{4}{3}$ for some angle $\theta$.