Question

if cosec theta +cot theta =5/2 then the value of tan theta is

Answer 1

Answer:tan a = 20/21

Step-by-step explanation:

cosec a + cot a = 5/2

Cosec^2 a + cot^2 a + 2cosec a. Cota =25/4

Using formula cosec^2 a= 1+cot^2 a,

Cot^2 a +1+ cot^2a + 2coseca. Cot a= 25/4

Rearranging:

2cot a(cot a + cosec a) = 21/4

Substituting cot a+ cosec a = 5/2,

2cot a. 5/2=21/4

Cot a = 21/20

Hence tan a = 20/21

Answer 2

Thus the value of $tan\alpha$ will be 20/21.

Given:

$\csc( \alpha ) + \cot( \alpha ) = \frac{5}{2}$

To find: We have to find the value of $tan\alpha$.

Solution:

We know that-

${ \csc }^{2} \alpha - { \cot }^{2} \alpha = 1 \\ ( \csc( \alpha ) + \cot( \alpha ) )( \csc( \alpha ) - \cot( \alpha ) ) = 1$

Putting the value of cosec alpha +cot alpha we get-

$\frac{5}{2} ( \csc( \alpha ) - \cot( \alpha ) ) = 1 \\ ( \csc( \alpha ) - \cot( \alpha ) ) = \frac{2}{5}$

Now putting the value of $csc\alpha$ in terms of $cot\alpha$ we get-

$\frac{5}{2} - \cot( \alpha ) - \cot( \alpha ) = \frac{2}{5} \\ - 2 \cot( \alpha ) = \frac{2}{5} - \frac{5}{2} \\ - 2 \cot( \alpha ) = \frac{ - 21}{10} \\ \cot( \alpha ) = \frac{21}{20}$

Now we know that $cot\alpha$ is an inverse of $tan\alpha$

Thus the value of $tan\alpha$ will be 20/21.