Question

How do you simplify \[\tan \left( 2x \right)\times \tan \left( 2x \right)\]?

Answer 1

Hint: In this problem, we have to simplify the given trigonometric expression. When we come to simplify the identities, we must use a combination of identities to reduce a complex expression to its simplest form. We can see that the expression has similar terms. So, we can just take the whole square for the term as the two terms are identical. We can then apply the whole square inside the terms.

Complete step-by-step answer:
We know that the given trigonometric expression is,
\[\tan \left( 2x \right)\times \tan \left( 2x \right)\]
We can see that the above expression contains products of two terms, where two terms are similar.
As the above expression has identical terms, we can now write the above expression in the form of whole square, we get
\[\Rightarrow {{\left( \tan 2x \right)}^{2}}\]
We can now apply the whole square inside the brackets to the tangent term, we get
\[\Rightarrow {{\tan }^{2}}2x\]
Therefore, the simplified form of the given expression \[\tan \left( 2x \right)\times \tan \left( 2x \right)\] is \[{{\tan }^{2}}2x\].

Note: Students make mistakes while analysing the question, where we can take the whole square only if we have two identical terms in the given expression. We should know that we can also apply the whole square which is outside the brackets, to the terms inside the brackets to get a simplified form. We should also concentrate in the symbol part as here we are given a product of two trigonometric terms. If we have given addition instead, we have to add the identical terms to get a simplified form.