Find: $\dfrac{{\cos {{45}^0}}} {{\sec {{30}^0} + \text{cosec}{{30}^0}}}$
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$\dfrac{2\sqrt 6}{\sqrt3 +1}$
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$\dfrac{\sqrt 2}{2\sqrt3 +2}$
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$\dfrac{2\sqrt 3}{\sqrt3 +1}$
- $ \dfrac{\sqrt{3}}{2\left( \sqrt{2}+\sqrt{6} \right)} $
cos 45 = 1/sqrt(2). sec 30 = 2/sqrt(3). cosec 30 = 2. Denominator = 2/sqrt(3) + 2 = (2 + 2sqrt(3))/sqrt(3). Expression = (1/sqrt(2)) / ((2 + 2sqrt(3))/sqrt(3)) = sqrt(3) / (sqrt(2) * 2 * (1 + sqrt(3))) = sqrt(3) / (2 * (sqrt(2) + sqrt(6))).
Substitute the standard trigonometric values into the expression to get cos 45 degrees divided by (sec 30 degrees + cosec 30 degrees). This becomes 1 divided by the square root of 2, all over (2 divided by the square root of 3) + 2. To simplify the denominator, write 2 as 2 multiplied by the square root of 3 divided by the square root of 3, giving a common denominator of the square root of 3. The denominator becomes the sum of 2 times the square root of 3 plus 2 times the square root of 3, which is 4 times the square root of 3 divided by the square root of 3. Dividing by this fraction gives the square root of 3 divided by the product of 4 times the square root of 2, which further simplifies by multiplying by the square root of 3 to yield the square root of 3 divided by 2 times the sum of the square root of 2 and the square root of 6.