Question

If cos A = 2 sin A, find cosec A.
12​

Answer 1

Given

$\bf \large \mapsto \: cos \theta = 2sin \theta$

To Find

$\bf \large \mapsto \: value \: of \: cosec \theta$

Concept used

$\begin{cases} \large \bf \: squaring \: both \: sides \\ \\ \large \bf \: {cos}^{2} \theta = 1 - {sin}^{2} \theta \\ \\ \large \bf \: cosec \theta = \dfrac{1}{sin \theta} \end{cases}$

Solution

write the equation which is given in question then we will observe the equation properly

$\rm \: cos \theta = 2sin \theta$

Now squaring both sides

$\rm \: {cos}^{2} \theta = 4 {sin}^{2} \theta$

Now using identity simplify it fastly

$\rm \:1 - {sin }^{2} \theta =4 {sin}^{2} \theta \\ \\ \implies \rm \: 1 = 5 {sin}^{2} \theta$

Now take 5 in LHS

$\rm \: {sin}^{2} \theta = \dfrac{1}{5} \\ \\ \implies \rm \: sin \theta = \pm \sqrt{ \frac{1}{5} } \\ \\ \implies \rm \: sin \theta = \pm \: \frac{1}{ \sqrt{5} }$

Using identity we get the answer easily (identity written in concept used)

$\rm \: \dfrac{1}{cosec \theta} = \pm\dfrac{1}{ \sqrt{5} } \\ \\ \bf \: reciprocal \\ \\ \implies \rm \underline{ \underline{ \boxed{ \bf \large \: cosec \theta = \pm \: \sqrt{5} }}}$

cosecQ is ±√5 is your answer to this question