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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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