Question

Find the value of $\cos ec31^\circ - \sec 59^\circ$

Answer 1

Here, we will find the value of the given expression of trigonometry. We will use the concept of trigonometric ratios of complementary angles to find the value of the expression. Trigonometry is used to find the relationships between the sides of a right angle triangle.

Formula Used:
Trigonometric ratio of complementary angles: $\cos ec\left( {90^\circ - \theta } \right) = \sec \theta$

Complete step-by-step answer:
We are given with an trigonometric expression $\cos ec31^\circ - \sec 59^\circ$.
We know that the angles are complementary to each other. So, adding the given angles, we get
$31^\circ + 59^\circ = 90^\circ$
Thus, we get $31^\circ = 90^\circ - 59^\circ$ …………………………………………………………………….$\left( 1 \right)$
By substituting equation $\left( 1 \right)$ in the given trigonometric expression, we get
$\Rightarrow \cos ec31^\circ - \sec 59^\circ = \cos ec\left( {90^\circ - 59^\circ } \right) - \sec 59^\circ$
By using the trigonometric ratio for complementary angles, we get
$\Rightarrow \cos ec31^\circ - \sec 59^\circ = \sec 59^\circ - \sec 59^\circ$
By subtracting the values, we get
$\Rightarrow \cos ec31^\circ - \sec 59^\circ = 0$
Therefore, the value of $\cos ec31^\circ - \sec 59^\circ$ is $0$.

Additional Information:
We know that the complementary angles are the set of angles which are complementary to each other and whose sum is equal to 90 degrees. We know that supplementary angles are the set of angles which are supplementary to each other and whose sum is equal to 180 degrees. Thus by using trigonometry of the complementary and supplementary angles, we will find the relations between the co-ratios in Trigonometry.

Note: We can also find the value in another method.
We know that the angles are complementary to each other.
So, we get $31^\circ + 59^\circ = 90^\circ$
Thus, we get $59^\circ = 90^\circ - 31^\circ$ …………………………………………………………………….$\left( 2 \right)$
By substituting equation $\left( 2 \right)$ in the given trigonometric expression, we get
$\Rightarrow \cos ec31^\circ - \sec 59^\circ = \cos ec31^\circ - \sec \left( {90^\circ - 31^\circ } \right)$
We know that the trigonometric ratio of complementary angles, $\sec \left( {90^\circ - \theta } \right) = \cos ec\theta$
By using the trigonometric ratio for complementary angles, we get
$\Rightarrow \cos ec31^\circ - \sec 59^\circ = \cos ec31^\circ - \cos ec31^\circ$
By subtracting the values, we get
$\Rightarrow \cos ec31^\circ - \sec 59^\circ = 0$
Therefore, the value of $\cos ec31^\circ - \sec 59^\circ$is $0$.