Question

If cosec A - cot A=4/5, then cosec A...​

Answer 1

cosec A - cot A = 4 / 5                                (i) eqn.

[Using identity :- $cosec^{2} A-cot^{2} A=1$ ]

[Also $a^{2}-b^{2}=(a-b)(a+b)$  ]

So, (cosec A + cot A )( cosec A - cot A ) = 1

(cosec A + cot A ) (4 / 5) = 1

cosec A + cot A = 5 / 4                                (ii) eqn.

From (i) & (ii) eqn.

cosec A - cot A = 4 / 5

cosec A + cot A = 5/ 4

So, 2 cosec A = 4 / 5 + 5 / 4

or,  2 cosec A = (16 + 25) / 20

or,  2 cosec A = 41 / 20

or,  cosec A = 41 / (20 × 2)

cosec A = 41 / 40

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Answer 2

Given,

$\[\cos ecA - \cot A = \frac{4}{5}\]$

Solution,

$\[\cos ecA - \cot A = \frac{4}{5}\]$

$\[\frac{1}{{\sin A}} - \frac{{\cos A}}{{\sin A}} = \frac{4}{5}\]$

$\[\frac{{1 - \cos A}}{{\sin A}} = \frac{4}{5}\]$

$\[ \Rightarrow \frac{{{{\cos }^2}\left( {\frac{A}{2}} \right) + {{\sin }^2}\left( {\frac{A}{2}} \right) - {{\cos }^2}\left( {\frac{A}{2}} \right) + si{n^2}\left( {\frac{A}{2}} \right)}}{{\sin A}} = \frac{4}{5}\]$

$\[ \Rightarrow \frac{{2si{n^2}\left( {\frac{A}{2}} \right)}}{{2\sin \left( {\frac{A}{2}} \right)\cos \left( {\frac{A}{2}} \right)}} = \frac{4}{5}\]$

$\[ \Rightarrow \tan \left( {\frac{A}{2}} \right) = \frac{4}{5}\]$------(1)

Observe the equation 1,

$\[\begin{array}{l}\sin \left( {\frac{A}{2}} \right) = \frac{4}{{\sqrt {41} }}\\\cos \left( {\frac{A}{2}} \right) = \frac{5}{{\sqrt {41} }}\end{array}\]$

Calculate the value of $\[\cos ecA\]$

$\[ = \cos ecA\]$

$\[ = \frac{1}{{\sin A}}\]$

$\[ = \frac{1}{{2\sin \left( {\frac{A}{2}} \right)\cos \left( {\frac{A}{2}} \right)}}\]$

$\[ = \frac{1}{{2 \times \frac{4}{{\sqrt {41} }} \times \frac{5}{{\sqrt {41} }}}}\]$

$\[ = \frac{{41}}{{2 \times 4 \times 5}}\]$

$\[ = \frac{{41}}{{40}}\]$

Hence the value of $\[\cos ecA\]$ is $\[ \frac{{41}}{{40}}\]$.