Question

If cos thetha =1/2 then find the value of 2 Sec thetha/1+tan² thetha​

Answer 1

Given

$\bf \mapsto \: cos \theta = \dfrac{1}{2}$

To Find

$\bf \mapsto \: value \: of \: \: \: \dfrac{2sec \theta}{1 + {tan}^{2} \theta }$

Concept used

$\bf \begin{cases} \bf {sec}^{2} \theta = 1 + {tan}^{2} \theta \: \: \: \: \boxed{1} \\ \\ \bf \: sec \theta = \dfrac{1}{cos \theta} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{2}\end{cases}$

Solution

We need to find the value of $\boxed{ \rm \: \frac{2sec \theta}{1 + {tan}^{2} \theta } }$

So we directly simplify it

$\rm \: \dfrac{2sec \theta}{1 + {tan}^{2} \theta } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf \: using \: identity \boxed{1} \\ \\ \rm \implies\dfrac{2sec \theta}{ {sec}^{2} \theta} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \rm \: \frac{2sec \theta}{sec \theta \times sec \theta} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf \: now \: cut \: sec \theta \\ \\ \implies \rm \: \frac{2}{sec \theta} \implies 2cos \theta \: \: \: \: \: \underline{\bf \: using \boxed{ 2} }$

We have the value of cos$\theta$

so we put in the formula easily

$\bf \: putting \: value \\ \\ 2 \times \frac{1}{2} \implies \: 1$

$\underline{ \boxed{ \bf \huge\frac{2sec \theta}{1 + {tan}^{2} \theta } = 1 }}$

It is the right way to solve this question fastly and easily