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Question: Simplify \(\dfrac{{{{\sin }^2}x - 1}}{{1 + {{\sin }^2}x}}\)?...

Simplify sin⁡2x−11+sin⁡2x\dfrac{{{{\sin }^2}x - 1}}{{1 + {{\sin }^2}x}}?

Explanation

Solution

In the given question, we've been given an expression. This expression has two trigonometric functions – one within the numerator and one within the denominator. We have to simplify the value of the trigonometric functions as a whole in the expression. 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 urge one account the entire expression.

Complete step by step solution:
The given expression is sin⁡2x−11+sin⁡2x\dfrac{{{{\sin }^2}x - 1}}{{1 + {{\sin }^2}x}}.
Trigonometric identities are equalities in mathematics that include trigonometric functions and are valid for each value of the variables occurring where both sides of the equality are defined. There are identities, geometrically, containing the functions of one or more angles. They are distinct from the identities of triangles, which are identities involving a triangle's angles and side lengths.
We know that,
sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1

Substitute it in the numerator of the expression,
⇒sin⁡2x−(sin⁡2x+cos⁡2x)1+sin⁡2x\Rightarrow \dfrac{{{{\sin }^2}x - \left( {{{\sin }^2}x + {{\cos }^2}x} \right)}}{{1 + {{\sin }^2}x}}
Open the brackets and change the sign accordingly,
⇒sin⁡2x−sin⁡2x−cos⁡2x1+sin⁡2x\Rightarrow \dfrac{{{{\sin }^2}x - {{\sin }^2}x - {{\cos }^2}x}}{{1 + {{\sin }^2}x}}
Simplify the terms,
⇒−cos⁡2x1+sin⁡2x\Rightarrow - \dfrac{{{{\cos }^2}x}}{{1 + {{\sin }^2}x}}

Hence, the simplified value of sin⁡2x−11+sin⁡2x\dfrac{{{{\sin }^2}x - 1}}{{1 + {{\sin }^2}x}} is −cos⁡2x1+sin⁡2x - \dfrac{{{{\cos }^2}x}}{{1 + {{\sin }^2}x}}.

Note: We got the solution to the present expression containing the 2 trigonometric functions by substituting the values of 1 because the combination of values of sine and cosine. Perhaps if we would like to simplify any expression containing the trigonometric functions, we will use these two to urge to the solution.
To do any quite simplification of trigonometric functions, we will just simplify them into sine and cosine then combine them then solve them. We just got to remember all the essential trigonometric identities.