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Find the value of \[\dfrac{{\sin {{53}^ \circ }}}{{\cos {{37}^ \circ }}} + 2\tan {45^ \circ } - \dfrac{{\cos ec{{60}^ \circ }}}{{\sec {{30}^ \circ }}}\]

Last updated date: 21st Jul 2024
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Hint:Consider the trigonometric functions to expand the equation with respect to its angles given. So that it will be cancelled easily and we will get the desired value.

Complete step by step answer:
To find the value of
\[\dfrac{{\sin {{53}^ \circ }}}{{\cos {{37}^ \circ }}} + 2\tan {45^ \circ } - \dfrac{{\cos ec{{60}^ \circ
}}}{{\sec {{30}^ \circ }}}\]
We must know the Trigonometric function chart which helps us to find the values of the function very easily.
Let us write the given terms
\[\dfrac{{\sin {{53}^ \circ }}}{{\cos {{37}^ \circ }}} + 2\tan {45^ \circ } - \dfrac{{\cos ec{{60}^ \circ
}}}{{\sec {{30}^ \circ }}}\]
As we cannot find the direct value of \[\sin {53^ \circ }\], hence let us split the value into \[\sin
\left( {90 - 37} \right)\], we know that the value of tan45 then the equation becomes
\[ = \dfrac{{\sin \left( {90 - 37} \right)}}{{\cos 37}} + 2\left( 1 \right) - \dfrac{{\cos ec\left( {90 - 30}
\right)}}{{\sec 30}}\]
\[ = \dfrac{{\cos 37}}{{\cos 37}} + 2 - \dfrac{{\sec 30}}{{\sec 30}}\]
\[ = 1 + 2 - 1\]
\[ = 2\]
Therefore, the value of
\[\dfrac{{\sin {{53}^ \circ }}}{{\cos {{37}^ \circ }}} + 2\tan {45^ \circ } - \dfrac{{\cos ec{{60}^ \circ
}}}{{\sec {{30}^ \circ }}} = 2\]

Additional information:
In trigonometry sin, cos and tan values are the primary functions we consider while solving
trigonometric problems. These trigonometry values are used to measure the angles and sides of a right-angle triangle. Apart from sine, cosine and tangent values, other values are cotangent, secant and cosecant.
When we find sin, cos and tan values for a triangle, we usually consider these angles: 0°, 30°, 45°, 60° and 90°. The trigonometric values are about the knowledge of standard angles for a given triangle as per the trigonometric ratios. Trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.
These angles can also be represented in the form of radians such as 0, π/6, π/4, π/3, and π/2. These angles are most commonly and frequently used in trigonometry.

Note: The key point to find the values of any trigonometric function is to note the chart of all functions as shown and calculates all the terms asked. And here are some of the formulas to be noted.
Tan θ = sin θ/cos θ
Cot θ = cos θ/sin θ
Sin θ = tan θ/sec θ
Cos θ = sin θ/tan θ
Sec θ = tan θ/sin θ
Cosec θ = sec θ/tan θ