Prove that frac{cot A-cos A}{cot A+cos A}=frac{operatorname{cosec} A-1}{operatorname{cosec} A+1}

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8. Introduction to Trigonometry

medium

Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$

Option A

Option B

Option C

Option D

Solution

$LHS =\frac{\cot A -\cos A }{\cot A +\cos A }=\frac{\frac{\cos A }{\sin A }-\cos A }{\frac{\cos A }{\sin A }+\cos A }$

$=\frac{\cos A\left(\frac{1}{\sin A}-1\right)}{\cos A\left(\frac{1}{\sin A}+1\right)}=\frac{\left(\frac{1}{\sin A}-1\right)}{\left(\frac{1}{\sin A}+1\right)}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}=R H S$

Standard 10

Mathematics

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta$

hard

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}$

medium

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 $A$.

medium

In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:

$(i)$ $\sin A, \cos A$

$(ii)$ $\sin C, \cos C$

medium

$\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}=$

easy

Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$

Solution

Similar Questions

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}$

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 $A$.

In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine: $(i)$ $\sin A, \cos A$ $(ii)$ $\sin C, \cos C$

$\frac{1-\tan ^{2} 45^{\circ}}{1+\tan ^{2} 45^{\circ}}=$

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Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{\sin \theta-2 \sin ^{3} \theta}{2 \cos ^{3} \theta-\cos \theta}=\tan \theta$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}$

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 $A$.

In $\triangle ABC ,$ right-angled at $B , AB =24 \,cm , BC =7 \,cm .$ Determine:

$(i)$ $\sin A, \cos A$

$(ii)$ $\sin C, \cos C$