Express trigonometric ratios sin A , sec A and tan A in terms of cot A .

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8. Introduction to Trigonometry

hard

Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.

Option A

Option B

Option C

Option D

Solution

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, $\sec ^{2} A=1+\tan ^{2} A$

$=1+\frac{1}{\cot ^{2} A}$

$=\frac{\cot ^{2} A+1}{\cot ^{2} A}$

$\sec A=\frac{\sqrt{\cot ^{2} A+1}}{\cot A}$

Standard 10

Mathematics

$\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}}=$

medium

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

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

hard

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

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

medium

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 $\theta$.

medium

If $\sin 3 A =\cos \left( A -26^{\circ}\right),$ where $3 A$ is an acute angle, find the value of $A= . . . . ^{\circ}$.

medium

Express the trigonometric ratios $\sin A , \sec A$ and $\tan A$ in terms of $\cot A$.

Solution

Similar Questions

$\frac{2 \tan 30^{\circ}}{1-\tan ^{2} 30^{\circ}}=$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

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

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 $\theta$.

If $\sin 3 A =\cos \left( A -26^{\circ}\right),$ where $3 A$ is an acute angle, find the value of $A= . . . . ^{\circ}$.

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

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$

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

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

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 $\theta$.