Question

If we have a trigonometric expression $3\tan A=4\sin A$, then find the relation between cosecA and cotA.

Answer 1

Hint: In the question, it is asked that we have to write $3\tan A=4\sin A$ in terms of cot A and cosec A. So, to do so we will use identities and properties of trigonometric ratios such as $\tan A=\dfrac{\sin A}{\cos A}$ and $\sin A=\dfrac{1}{\cos ecA}$ so as to obtain the $3\tan A=4\sin A$ in terms of ${\cot A}$ and $\cos ecA$.

Complete step-by-step solution:
We know that ${\sin A}$, ${\cos A}$, ${\tan A}$, ${\cot A}$, $\sec A$, and $\cos ecA$ are trigonometric functions, where A is the angle made by the hypotenuse with the base of the triangle.
Now, in the question, it is given that $3\tan A=4\sin A$.
Now, also we know that tan A equals the ratio of the sine function and cos A function that is $\tan A=\dfrac{\sin A}{\cos A}$.
And, also ${\sin A}$ is equals to the reciprocal of trigonometric function cosec A that is $\sin A=\dfrac{1}{\cos ecA}$.
so, we can write $3\tan A=4\sin A$ as,
$3\tan A=4\sin A$.
$3\dfrac{\sin A}{\cos A}=4\dfrac{1}{\cos ecA}$
Taking $\cos ecA$ from the denominator of the right-hand side to the numerator of the left-hand side, ${\sin A}$ from the numerator of the left-hand side to the denominator of the right-hand side, and $\cos A$ from the denominator of the left-hand side to the numerator of the right-hand side, using cross multiplication, we get
$3\cos ecA=4\dfrac{\cos A}{\sin A}$……..( i )
Now, also ${\cot A}$ equals to the reciprocal of the trigonometric function ${\tan A}$ that is $\cot A=\dfrac{1}{\tan A}$
But, also as we discussed above that $\tan A=\dfrac{\sin A}{\cos A}$,
So, $\cot A=\dfrac{1}{\dfrac{\sin A}{\cos A}}$
On simplifying, we get
$\cot A=\dfrac{\cos A}{\sin A}$
Now in equation ( i ), we can write $4\dfrac{\sin A}{\cos A}$ as $4{\cot A}$
Thus, we have $3\cos ecA=4\cot A$.
Hence, the relation between cosec A and cot A for $3\tan A=4\sin A$ is equal to $3\cos ecA=4\cot A$.

Note: One must know the relation between trigonometric functions such as $\tan A=\dfrac{\sin A}{\cos A}$, $\cot A=\dfrac{\cos A}{\sin A}$, also we can use some direct trigonometric substitution such as $\sin A=\dfrac{1}{\cos ecA}$, $\cot A=\dfrac{1}{\tan A}$ and on the conversion of tan into cot and sin into cosec and as we have to find the final answer in terms of cosecA and cotA so it will solve question in a better and faster way. While solving the question always use the most appropriate substitution of trigonometric relation which directly leads to results. Try not to make any calculation mistakes.