Find the value of $\frac{1}{tanA}+\frac{sinA}{( 1+cosA)}$, if $cosec A=2$.

Given: $cosec A=2$.

To do: To find the value of $\frac{1}{tanA}+\frac{sinA}{( 1+cosA)}$.

Solution:

$cosecA=2$

$\Rightarrow sinA = \frac{1}{2}$ ...... $( i)$

Therefore, $cosA=\sqrt{( 1-sin²A)}$

$\Rightarrow  \sqrt{( 1-\frac{1}{4})}$

$\Rightarrow \sqrt{( \frac{3}{4})}$

$\Rightarrow  \frac{\sqrt{3}}{2}$ ...... $( ii)$

So, $tanA=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

$=( \frac{1}{2})\times ( \frac{2}{\sqrt{3}})$

$=\frac{1}{\sqrt{3}}$ ....... $( iii)$

Now in $\frac{1}{tanA}+\frac{sinA}{( 1+cosA)}$

By $( i),\ ( ii)$ & $( iii)$, put the value

$=\frac{1}{\frac{1}{\sqrt{3}}}+\frac{\frac{1}{2}}{1+\frac{\sqrt{3}}{2}}$

$=\sqrt{3}+\frac{( \frac{1}{2})}{ ( \frac{2+\sqrt{3}}{2})}$

$= \sqrt{3}+\frac{1}{(2+\sqrt{3})}$

$= \frac{( 2\sqrt{3}+3+1)}{( 2+\sqrt{3})}$

$=2\frac{( \sqrt{3}+2)}{( \sqrt{3}+2)}$

$= 2$

Updated on: 10-Oct-2022

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