Evaluate \[\cos {{48}^{\circ }}-\sin {{42}^{\circ }}\] ?

Hint: For solving this problem we need to have a clear understanding about trigonometry and the basic formulas of trigonometry. We also need to know the relation between the sin and cos functions. By converting the value of cos function into sine function by using appropriate formula, we can easily get the answer to the above problem.

Complete step by step answer:
Trigonometry is a branch of mathematics that deals with triangles. Trigonometry is also known as the study of relationships between lengths and angles of triangles. There is an enormous number of uses of trigonometry and its formulae. In Trigonometry, different types of problems can be solved using trigonometry formulas. These problems may include trigonometric ratios (sin, cos, tan, sec, cosec and cot), Pythagorean identities, product identities, etc. Trigonometric Identities are formulas that involve Trigonometric functions. These identities are true for all values of the variables. Trigonometric Ratio is known for the relationship between the measurement of the angles and the length of the sides of the right triangle. Basically, they are the ratios of sides of a right-angle triangle. For solving this problem, we need the concept and formula for co-function or periodic identities.
To find \[\cos {{48}^{\circ }}-\sin {{42}^{\circ }}\] , we know that
\[\cos \left( {{90}^{\circ }}-\theta \right)=\sin \theta \ldots .\text{ }\left( 1 \right)\]
So, we can write \[\cos {{48}^{\circ }}\] as
\[\begin{align}
  & \Rightarrow \cos {{48}^{\circ }}=\cos \left( {{90}^{\circ }}-{{42}^{\circ }} \right) \\
 & \Rightarrow \cos {{48}^{\circ }}=\sin {{42}^{\circ }}....\left( 2 \right) \\
\end{align}\]
From equation $\left( 2 \right)$ we can write the given expression as
\[\sin {{42}^{\circ }}-\sin {{42}^{\circ }}=0\]
Thus, the answer to the problem is zero.

Note: These types of problems are pretty easy to solve but only if we have the concept and the knowledge of the formulas of co-function or periodic identities. A slight error in the transformation of cos angle into sin angle can lead to a totally different answer. Hence, one must memorize the formulas and apply them correctly.