Question

Evaluate \[\dfrac{{\tan {{45}^0}}}{{\csc {{30}^0}}} + \dfrac{{\sec {{60}^0}}}{{\cot {{45}^0}}} - \dfrac{{5\sin {{90}^0}}}{{2\cos {0^0}}}\]

Answer 1

Hint: Before evaluating this problem observe that, in these types of questions we need to check whether the angle is in standard form or not. We can observe that all the angels are in standard form, meaning we know these values. Directly substitute and simplify. If not, we use the trigonometric ratios to solve these questions that is by expressing trigonometric functions in terms of their complements. We can also express the trigonometric functions in terms of their supplements.

Complete step-by-step answer:
There are six trigonometric ratios, namely
Sine, cosine, tangent, cosecant, secant and cotangent.
Also remember, by convection, positive angles are measured in the anti-clockwise direction starting from the positive x axis. Domain ranges between -1 and 1 only. We use right angle triangles while defining the trigonometric relations. That is,\[\theta = {90^0}\]
Given,
\[\dfrac{{\tan {{45}^0}}}{{\csc {{30}^0}}} + \dfrac{{\sec {{60}^0}}}{{\cot {{45}^0}}} - \dfrac{{5\sin {{90}^0}}}{{2\cos {0^0}}}\]
We know the values of these individually,
$\Rightarrow$ \[\tan {45^0} = 1\],\[\csc {30^0} = 2\],\[\cot {45^0} = 1\], \[\sin {90^0} = 1\] and \[\cos {0^0} = 1\].
We know these values by standard trigonometric ratios. Which we need to remember.
Substituting these values in above equation we get,
\[ = \dfrac{1}{2} + \dfrac{2}{1} - \dfrac{{5}}{{2}}\]
Using simple multiplication and division.
\[ = \dfrac{1}{2} + 2 - \dfrac{5}{2}\]
Taking L.C.M. and simplifying we get,
\[ = \dfrac{{1 + 4 - 5}}{2}\]
\[ = \dfrac{{5 - 5}}{2}\] [Using simple subtraction and division.]
5 and 5 gets cancels out, we get
\[ = 0\]
Thus we get, \[\dfrac{{\tan {{45}^0}}}{{\csc {{30}^0}}} + \dfrac{{\sec {{60}^0}}}{{\cot {{45}^0}}} - \dfrac{{5\sin {{90}^0}}}{{2\cos {0^0}}} = 0\]
So, the correct answer is “0”.

Note: We can see that \[\tan \theta \] and \[\cot \theta \] are reciprocal and hence have the same value at the angle \[{45^0}\]. Here, the question is straightforward so no need to think about it, we can substitute the known values and evaluate. Always remember the trigonometric ratios value. If we know these we can easily solve these types of questions.