Question

Find the value of \[\dfrac{{\cos ec{{13}^o}}}{{\sec {{77}^o}}} - \dfrac{{\cot {{20}^o}}}{{\tan {{70}^o}}}\]

Answer 1

Hint: Here, we will use two different basic trigonometry formulas and convert cosec to sec and cot to tan respectively. We will then evaluate both to get the desired results.

Formula used:
 Here, we will use the formula \[\cos ec\left( {90 - \theta } \right) = \sec \theta \] and \[\cot \left( {90 - \theta } \right) = \tan \theta \] .

Complete step-by-step answer:
 We will use basic trigonometry formulas to evaluate this trigonometric equation.
\[\dfrac{{\cos ec{{13}^o}}}{{\sec {{77}^o}}} - \dfrac{{\cot {{20}^o}}}{{\tan {{70}^o}}}\]
As both of the numerator and denominator have different trigonometric functions that is cosec and sec. We will use trigonometric formula \[\cos ec\left( {90 - \theta } \right) = \sec \theta \] to convert cosec to sec.
\[\dfrac{{\cos ec\left( {{{90}^o} - {{13}^o}} \right)}}{{\sec {{77}^o}}} - \dfrac{{\cot \left( {{{20}^o}} \right)}}{{\tan {{70}^o}}}\] = equation 1
By using the formula, \[\cos ec\left( {90 - \theta } \right) = \sec \theta \]
We can calculate the \[\cos ec\left( {{{90}^o} - {{13}^o}} \right)\] which is equal to \[\sec {77^o}\] .
Using the values we calculated, we will put them in equation 1.
As both of the numerator and denominator have different trigonometric functions that is cot and tan. We will use trigonometric formula \[\cot \left( {90 - \theta } \right) = \tan \theta \] to convert cot to tan.
\[\dfrac{{sec{{77}^o}}}{{\sec {{77}^o}}} - \dfrac{{\cot \left( {{{90}^o} - {{20}^o}} \right)}}{{\tan {{70}^o}}}\] = equation 2
By using the formula, \[\cot \left( {90 - \theta } \right) = \tan \theta \]
We can calculate the \[\cot \left( {{{90}^0} - {{20}^o}} \right)\] which is equal to \[\tan {70^0}\] .
Using the values we calculated, we will put them in equation 2.
We get, \[\dfrac{{\sec {{77}^o}}}{{\sec {{77}^o}}} - \dfrac{{\tan {{70}^o}}}{{\tan {{70}^o}}}\]
Canceling all the numerator and the denominator with each other leaves us with \[1 - 1\]
That eventually leads to \[0\]
Therefore we can conclude that, \[\dfrac{{\cos ec{{13}^o}}}{{\sec {{77}^o}}} - \dfrac{{\cot {{20}^o}}}{{\tan {{70}^o}}} = 0\]

Additional information:
We use multiple trigonometric formulas to convert these types of questions into their simplest forms. Some of these formulas commonly used are \[\cot \left( {90 - \theta } \right) = \tan \theta \] to convert cot into tan, \[\cos ec\left( {90 - \theta } \right) = \sec \theta \] to convert cosec into sec. and \[\sin \left( {90 - \theta } \right) = \cos \theta \] to convert sin to cos.

Note: In these types of questions, do not forget to use the trigonometric formulas as it makes the question simplest and make it easy to evaluate. Similarly, most of the trigonometric problems can be solved using basic trigonometric operations and formulas.