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Question: How do you simplify the expression \[\dfrac{{{{\sin }^2}x}}{{1 + \cos x}}\]?...

How do you simplify the expression sin⁡2x1+cos⁡x\dfrac{{{{\sin }^2}x}}{{1 + \cos x}}?

Explanation

Solution

Here, we will use the trigonometric identity to convert the sine function in the numerator to the cosine function. Then by using a suitable algebraic identity, we will simplify the expression to make the numerator the same as the denominator. Thus, the simplified expression is the required answer.

Formula used:
We will use the following formula:
Trigonometric Identity: sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1
The difference between the square of the numbers is given by an algebraic identity a2−b2=(a+b)(a−b){a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)

Complete Step by Step Solution:
We are given the expression sin⁡2x1+cos⁡x\dfrac{{{{\sin }^2}x}}{{1 + \cos x}}.
Let f(x)f\left( x \right) be the given expression.
f(x)=sin⁡2x1+cos⁡xf\left( x \right) = \dfrac{{{{\sin }^2}x}}{{1 + \cos x}}
We know that the trigonometric identity: sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1
Now, by rewriting the equation, we get
⇒sin⁡2x=1−cos⁡2x\Rightarrow {\sin ^2}x = 1 - {\cos ^2}x
By substituting the rewritten trigonometric identity in the given expression, we get
f(x)=1−cos⁡2x1+cos⁡xf\left( x \right) = \dfrac{{1 - {{\cos }^2}x}}{{1 + \cos x}}.
The difference between the square of the numbers is given by an algebraic identity a2−b2=(a+b)(a−b){a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)
By using the algebraic identity, we get
⇒f(x)=(1−cos⁡x)(1+cos⁡x)1+cos⁡x\Rightarrow f\left( x \right) = \dfrac{{\left( {1 - \cos x} \right)\left( {1 + \cos x} \right)}}{{1 + \cos x}}
By canceling the terms in the numerator and in the denominator, we get
⇒f(x)=1−cos⁡x\Rightarrow f\left( x \right) = 1 - \cos x

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

Note: We know that we have many trigonometric identities that are related to all the other trigonometric equations. We should remember that the trigonometric ratio and the co-trigonometric ratio is always reciprocal to each other. Trigonometric ratios are used to find the relationships between the sides of a right-angle triangle. We know that the Trigonometric equation is defined as an equation involving the trigonometric ratios. Trigonometric identity is an equation that is always true. An algebraic equation is defined as the combinations of variables and constants. The algebraic equation which is valid for all the variables in an equation is called the algebraic identity.