Prove that: \[(\text{sec}A+\cos A)(\sec A-\cos A)={{\tan }^{2}}A+{{\sin }^{2}}A\]

Hint: We will begin with the left hand side of the expression and then we will transform the two terms into one term using \[(a-b)(a+b)={{a}^{2}}-{{b}^{2}}\]. Then we will convert \[{{\sec }^{2}}A\] into \[{{\tan }^{2}}A\] using the formula \[{{\sec }^{2}}A=1+{{\tan }^{2}}A\] and \[{{\cos }^{2}}A\] into \[{{\sin }^{2}}A\] using the formula \[{{\cos }^{2}}A=1-{{\sin }^{2}}A\]. Then we will cancel the similar terms to get the expression same as the right hand side of the given expression.

Complete step-by-step answer:

It is mentioned in the question that \[(\text{sec}A+\cos A)(\sec A-\cos A)={{\tan }^{2}}A+{{\sin }^{2}}A........(1)\]

Now beginning with the left hand side of the equation (1) we get,

\[\Rightarrow (\text{sec}A+\cos A)(\sec A-\cos A)........(2)\]

We know that \[(a-b)(a+b)={{a}^{2}}-{{b}^{2}}\]. So here assuming a as secA and b as cosA and using this formula in equation (2) we get,

\[\Rightarrow \text{se}{{\text{c}}^{2}}A-{{\cos }^{2}}A........(3)\]

Now we know that \[{{\cos }^{2}}A=1-{{\sin }^{2}}A\] and \[{{\sec }^{2}}A=1+{{\tan }^{2}}A\].

So applying this in equation (3) we get,

\[\Rightarrow (1+{{\tan }^{2}}A)-(1-{{\sin }^{2}}A)........(4)\]

Now opening the brackets and cancelling similar terms in equation (4) we get,

\[\Rightarrow {{\tan }^{2}}A+{{\sin }^{2}}A........(5)\]

Hence from equation (5) we can say that the left hand side is equal to the right hand side in equation (1). Hence we have proved the given expression.

Note: In trigonometry remembering the formulas and the identities is very important because then it becomes easy. We may get confused about how to proceed further after equation (3) but here the key is to substitute \[1-{{\sin }^{2}}A\] in place of \[{{\cos }^{2}}A\] and \[1+{{\tan }^{2}}A\] in place of \[{{\sec }^{2}}A\]. Also in a hurry we can make a mistake while opening the bracket in equation (4) as we may keep \[-{{\sin }^{2}}A\] as \[-{{\sin }^{2}}A\] without changing the minus sign and hence we need to be careful while doing this step.