Could someone please help me prove this identity? $\dfrac{1}{{\sec A - 1}} + \dfrac{1}{{\sec A + 1}} = 2\cot A\cos ecA$

Hint: In this question we need to use some identities of trigonometry to solve this question. The identities we need in this question are $1 + {\tan ^2}x = {\sec ^2}x$ , $\sec A = \dfrac{1}{{\cos A}}$, $\cot A = \dfrac{{\cos A}}{{\sin A}}$and $\cos ecA = \dfrac{1}{{\sin A}}$. They need to be used in this question wherever we find their use while solving the question.

Complete step by step answer:
In the above question, let’s take L.H.S
$ \Rightarrow \dfrac{1}{{\sec A + 1}} + \dfrac{1}{{\sec A - 1}}$
Now let’s take LCM and solve this question further.
$ \Rightarrow \dfrac{{\sec A - 1 + \sec A + 1}}{{\left( {\sec A + 1} \right)\left( {\sec A - 1} \right)}}$
Using identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ and on further simplification, we get
$ \Rightarrow \dfrac{{2\sec A}}{{\left( {{{\sec }^2}A - 1} \right)}}$
Now using the identity $1 + {\tan ^2}x = {\sec ^2}x$ and on further simplification, we get
$ \Rightarrow \dfrac{{2\sec A}}{{{{\tan }^2}A}}$
Now, we will use the identity $\tan A = \dfrac{{\sin A}}{{\cos A}}$.
$ \Rightarrow \dfrac{{2\sec A}}{{\dfrac{{{{\sin }^2}A}}{{{{\cos }^2}A}}}}$
Now we will use the identity which will convert $\sec A$ into $\cos A$as $\sec A = \dfrac{1}{{\cos A}}$ and taking the ${\cos ^2}A$ to numerator.
$ \Rightarrow \dfrac{2}{{\cos A}}\dfrac{{{{\cos }^2}A}}{{{{\sin }^2}A}}$
Now cancelling the $\cos A$
$ \Rightarrow 2.\dfrac{{\cos A}}{{\sin A}}.\dfrac{1}{{\sin A}}$
Now we will use the identities $\cot A = \dfrac{{\cos A}}{{\sin A}}$ and $\cos ecA = \dfrac{1}{{\sin A}}$.
$ \Rightarrow 2\cot A\cos ecA$
Also, we have $RHS = 2\cot A\cos ecA$
Therefore, $LHS = RHS$
Hence proved.

Additional information: Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Identity inequalities which are true for every value occurring on both sides of an equation. Geometrically, these identities involve certain functions of one or more angles. There are various distinct identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.

Note: In this question we have used a lot of identities. Therefore, we need to remember all the identities as trigonometry is nothing without identities. Most of the questions we can just do by simply applying identities.