Evaluate: cos 45 / (sec 30 + cosec 30)

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Q) Evaluate: $\frac{\cos 45}{(\sec 30 + \cos ec 30)}$

Ans: We need to find the value of: $\frac{\cos 45}{(\sec 30 + \cos ec 30)}$

We know that cos 45 = $\frac{1}{\sqrt 2}$ , sec 30 = $\frac{2}{\sqrt 3}$ , cosec 30 = 2

by substituting these values in the given expression, we get:

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

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

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

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

= $\frac{\sqrt 3}{2 \sqrt 6 + 2 \sqrt 2)}$ …. Answer

or to further simplify, we can multiply and divide the expression by (√3 – 1):

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

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

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

= $\frac{\sqrt 3 (\sqrt 3 - 1)}{4 \sqrt 2}$

= $\frac{3 - \sqrt 3}{4 \sqrt 2}$

multiply and divide the expression by √2:

= $\frac{3 - \sqrt 3}{4 \sqrt 2} \times \frac{\sqrt 2}{\sqrt 2}$

= $\frac{\sqrt 2(3 - \sqrt 3)}{4 \times 2}$

= $\frac{1}{8} (3\sqrt 2 - \sqrt 6)$ ….. Answer

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