Simplify The Expression:${ \tan X \left(\sin^2 X + 1 - \sin X\right) }$

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Introduction

In this article, we will delve into the world of trigonometry and explore the process of simplifying a given expression involving tangent and sine functions. The expression we will be working with is tan⁡x(sin⁡2x+1−sin⁡x)\tan x \left(\sin^2 x + 1 - \sin x\right). Our goal is to simplify this expression and provide a clear understanding of the steps involved.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its components. The expression consists of two main parts: tan⁡x\tan x and (sin⁡2x+1−sin⁡x)\left(\sin^2 x + 1 - \sin x\right). The tangent function is defined as the ratio of the sine and cosine functions, i.e., tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}. The expression inside the parentheses involves the sine function and its square.

Step 1: Simplify the Expression Inside the Parentheses

To simplify the expression inside the parentheses, we can start by factoring out the common term sin⁡x\sin x. This gives us:

sin⁡2x+1−sin⁡x=sin⁡x(sin⁡x−1)+1\sin^2 x + 1 - \sin x = \sin x(\sin x - 1) + 1

Now, we can see that the expression inside the parentheses can be rewritten as a product of two terms: sin⁡x\sin x and (sin⁡x−1)(\sin x - 1). We can then use the distributive property to expand the expression:

sin⁡x(sin⁡x−1)+1=sin⁡2x−sin⁡x+1\sin x(\sin x - 1) + 1 = \sin^2 x - \sin x + 1

Step 2: Simplify the Expression Using Trigonometric Identities

Now that we have simplified the expression inside the parentheses, we can focus on simplifying the entire expression. We can start by using the trigonometric identity tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} to rewrite the expression:

tan⁡x(sin⁡2x+1−sin⁡x)=tan⁡x(sin⁡2x−sin⁡x+1)\tan x \left(\sin^2 x + 1 - \sin x\right) = \tan x \left(\sin^2 x - \sin x + 1\right)

Next, we can use the fact that tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x} to rewrite the expression as:

tan⁡x(sin⁡2x−sin⁡x+1)=sin⁡xcos⁡x(sin⁡2x−sin⁡x+1)\tan x \left(\sin^2 x - \sin x + 1\right) = \frac{\sin x}{\cos x} \left(\sin^2 x - \sin x + 1\right)

Step 3: Simplify the Expression Using Algebraic Manipulations

Now that we have rewritten the expression using trigonometric identities, we can focus on simplifying it using algebraic manipulations. We can start by multiplying the numerator and denominator of the expression by cos⁡x\cos x:

sin⁡xcos⁡x(sin⁡2x−sin⁡x+1)=sin⁡x(sin⁡2x−sin⁡x+1)cos⁡x\frac{\sin x}{\cos x} \left(\sin^2 x - \sin x + 1\right) = \frac{\sin x \left(\sin^2 x - \sin x + 1\right)}{\cos x}

Next, we can use the distributive property to expand the numerator:

sin⁡x(sin⁡2x−sin⁡x+1)cos⁡x=sin⁡3x−sin⁡2x+sin⁡xcos⁡x\frac{\sin x \left(\sin^2 x - \sin x + 1\right)}{\cos x} = \frac{\sin^3 x - \sin^2 x + \sin x}{\cos x}

Step 4: Simplify the Expression Using Trigonometric Identities

Now that we have simplified the expression using algebraic manipulations, we can focus on simplifying it using trigonometric identities. We can start by using the trigonometric identity sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1 to rewrite the expression:

sin⁡3x−sin⁡2x+sin⁡xcos⁡x=sin⁡x(sin⁡2x−sin⁡x+1)cos⁡x\frac{\sin^3 x - \sin^2 x + \sin x}{\cos x} = \frac{\sin x \left(\sin^2 x - \sin x + 1\right)}{\cos x}

Next, we can use the fact that sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1 to rewrite the expression as:

sin⁡x(sin⁡2x−sin⁡x+1)cos⁡x=sin⁡x(1−cos⁡2x−sin⁡x+1)cos⁡x\frac{\sin x \left(\sin^2 x - \sin x + 1\right)}{\cos x} = \frac{\sin x \left(1 - \cos^2 x - \sin x + 1\right)}{\cos x}

Step 5: Simplify the Expression Using Algebraic Manipulations

Now that we have rewritten the expression using trigonometric identities, we can focus on simplifying it using algebraic manipulations. We can start by multiplying the numerator and denominator of the expression by cos⁡x\cos x:

sin⁡x(1−cos⁡2x−sin⁡x+1)cos⁡x=sin⁡x(2−cos⁡2x−sin⁡x)cos⁡x\frac{\sin x \left(1 - \cos^2 x - \sin x + 1\right)}{\cos x} = \frac{\sin x \left(2 - \cos^2 x - \sin x\right)}{\cos x}

Next, we can use the distributive property to expand the numerator:

sin⁡x(2−cos⁡2x−sin⁡x)cos⁡x=2sin⁡x−sin⁡xcos⁡2x−sin⁡2xcos⁡x\frac{\sin x \left(2 - \cos^2 x - \sin x\right)}{\cos x} = \frac{2 \sin x - \sin x \cos^2 x - \sin^2 x}{\cos x}

Conclusion

In this article, we have simplified the expression tan⁡x(sin⁡2x+1−sin⁡x)\tan x \left(\sin^2 x + 1 - \sin x\right) using a combination of trigonometric identities and algebraic manipulations. We have shown that the expression can be rewritten as 2sin⁡x−sin⁡xcos⁡2x−sin⁡2xcos⁡x\frac{2 \sin x - \sin x \cos^2 x - \sin^2 x}{\cos x}. This simplified expression provides a clear understanding of the relationship between the tangent and sine functions.

Final Answer

The final answer is 2sin⁡x−sin⁡xcos⁡2x−sin⁡2xcos⁡x\boxed{\frac{2 \sin x - \sin x \cos^2 x - \sin^2 x}{\cos x}}.