\(\frac{cos 45°}{sec30° + cosec 30°}\)

We know that,

cos 45° = 1/√2

sec 30° = 2/√3

cosec 30° = 2

Substitute the values, we get

\(\frac{cos 45°}{sec30° + cosec 30°}\) = \(\frac{\frac{1}{\sqrt 2}}{\frac{2}{\sqrt 3} + 2}\) = \(\frac{\frac{1}{\sqrt 2}}{\frac{2+2 \sqrt 3}{\sqrt 3}}\)

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

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

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

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

Now, rationalize the terms we get,

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

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

Now, multiply both the numerator and denominator by √2 , we get

= \(\frac{3-\sqrt 3}{2 \sqrt 2 (2)} \times \frac{\sqrt 2}{\sqrt 2} \) = \(\frac{3\sqrt 2 - \sqrt 3 \sqrt 2}{8} \)

= \(\frac{3\sqrt 2 - \sqrt 6}{8} \)

Therefore, cos 45°/(sec 30°+cosec 30°)

= \(\frac{3\sqrt 2 - \sqrt 6}{8} \)