Find \[\left( \sec A+\tan A-1 \right)\left( \sec A-\tan A+1 \right)=\]
(a) \[2\sin A\]
(b) \[2\cos A\]
(c) \[2\sec A\]
(d) \[2\tan A\]

Hint: In this question, we first need to know about some of the basic definitions of trigonometry. Then expand the given equation by multiplication and then use the trigonometric identities to simplify it further.
\[{{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1\]

Complete step by step solution:
Now, by expanding the given equation we get,
\[\begin{align}
  & \Rightarrow \left( \sec A+\tan A-1 \right)\left( \sec A-\tan A+1 \right) \\
 & \Rightarrow \left( \sec A+\tan A-1 \right)\left( \sec A-\left( \tan A-1 \right) \right) \\
\end{align}\]
\[\left[ \because \left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}} \right]\]
\[\begin{align}
  & \Rightarrow {{\sec }^{2}}A-{{\left( \tan A-1 \right)}^{2}} \\
 & \Rightarrow {{\sec }^{2}}A-\left( {{\tan }^{2}}A+1-2\tan A \right) \\
\end{align}\]
\[\Rightarrow {{\sec }^{2}}A-{{\tan }^{2}}A-1+2\tan A\]
\[\left[ \because {{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1 \right]\]
\[\begin{align}
  & \Rightarrow 1-1+2\tan A \\
 & \Rightarrow 2\tan A \\
\end{align}\]
Hence, the correct option is (d).

Note: Instead of expanding the given equation in terms of secant and tangent function and then applying the corresponding identity we can also convert the secant and tangent functions into sine and cosine functions using the respective identities and then expand it to simplify using the corresponding identity. Both the methods give the same result.
\[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
While simplifying the equation we need to be careful and apply the corresponding values accordingly because neglecting any of the terms changes the result of the equation completely.