Question

How do you simplify \[\left( {1 - {{\cos }^2}x} \right)\csc x\]?

Answer 1

Hint: We know that all the trigonometric functions can be represented in a combination of sine and cosine and that is how we will simplify each of the trigonometric functions, and then combine them both to get a single answer for the whole expression.

Complete step-by-step answer:
The given expression is \[p = \left( {1 - {{\cos }^2}x} \right)\csc x\].
Now, we know that:
\[1 - {\cos ^2}x = {\sin ^2}x\]
and
\[\csc x = \dfrac{1}{{\sin x}}\]
Hence, \[p = {\sin ^2}x \times \dfrac{1}{{\sin x}}\]
Simplifying them,
\[p = {{{{\sin }^2}x}}\sin x \times \dfrac{1}{{{{\sin x}}}} = \sin x\]
Hence, \[\left( {1 - {{\cos }^2}x} \right)\csc x = \sin x\]

Additional Information:
We got the answer to this expression containing the two trigonometric functions by substituting the values of the secant and tangent as the combination of values of sine and cosine. Perhaps if we want to simplify any expression containing the trigonometric functions, we can use these two to get to the answer.

Note: In the given question, we had to simplify the value of an expression containing two trigonometric functions. To do any kind of simplification of trigonometric functions, we can just simplify them into sine and cosine and then combine them and then solve them. We just need to remember all the basic trigonometric identities.