简化三角数表达式

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How could you write the trigonometric function $\begin{array}{r}\cos \theta +\cos \theta ({\tan }^2\theta )\end{array}$ more simply?
::如何更简单地写入三角函数 ?

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Simplifying Trigonometric Expressions
::简化三角数表达式

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Now that you are more familiar with trig identities, we can use them to simplify expressions. Remember, that you can use any of the following identities.
::现在您更熟悉三类身份, 我们可以使用它们来简化表达式。 记住, 您可以使用以下任何一种身份 。

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: $\begin{array}{r}\csc \theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\end{array}$ and $\begin{array}{r}\cot \theta =\frac{1}{\tan \theta }\end{array}$
:csc1sin,sec1cos,和cot1tan)

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Tangent and Cotangent Identities: $\begin{array}{r}\tan \theta =\frac{\sin \theta }{\cos \theta }\end{array}$ and $\begin{array}{r}\cot \theta =\frac{\cos \theta }{\sin \theta }\end{array}$
::相近和相切的特征:tansincos和cotcossin

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: $\begin{array}{r}{\sin }^2\theta +{\cos }^2\theta =1,1+{\tan }^2\theta ={\sec }^2\theta ,\end{array}$ and $\begin{array}{r}1+{\cot }^2\theta ={\csc }^2\theta \end{array}$
::1,1+tan2sec2和1+cot2csc2

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Cofunction Identities: $\begin{array}{r}\sin \left(\frac{\pi }{2}-\theta \right)=\cos \theta ,\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta ,\end{array}$ and $\begin{array}{r}\tan \left(\frac{\pi }{2}-\theta \right)=\cot \theta \end{array}$
::共认人:sin(2) =cos,cos(2) =sin, 和tan(2) =cot

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Negative Angle Identities: $\begin{array}{r}\sin (-\theta )=-\sin \theta ,\cos (-\theta )=\cos \theta ,\end{array}$ and $\begin{array}{r}\tan (-\theta )=-\tan \theta \end{array}$
::负角识别度: sin sin, cos =cos, 和 tan tan

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Let's simplify the following expressions.
::让我们简化以下表达式。

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1. $\begin{array}{r}\frac{\sec x}{\sec x-1}\end{array}$
   ::秒 *%xsec*x- 1

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When simplifying trigonometric expressions, one approach is to change everything into sine or cosine. First, we can change secant to cosine using the Reciprocal Identity .
::当简化三角表达式时,一种方法是将一切改变为正弦或正弦。 首先,我们可以用“对等身份”来改变“分弦”。

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::秒 *%ssec*x - 11cos *x1cos*x1cos*x-1

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Now, combine the denominator into one fraction by multiplying 1 by $\begin{array}{r}\frac{\cos x}{\cos x}\end{array}$ .
::现在,将分母结合成一个分数, 乘以 1 乘以 cos xxcos x 。

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::1cos*x1cos*x1cos*x1cos*x1cos*x1cos*xxx1cos*xx1cos*xx

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Change this problem into a division problem and simplify.
::将这个问题变成分裂问题和简化。

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::1cos*x1 -cos*xx*x*x*x*1x*x*1 -cos*x*x*x*x1x*x*x*x*x*_cos*x*x*x*1_cos*x11_cos*x*x

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2. $\begin{array}{r}\frac{{\sin }^{4}x-{\cos }^{4}x}{{\sin }^{2}x-{\cos }^{2}x}\end{array}$
   ::x-cos4xsin2x-cos2xxxx2xxxxxxxxx

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With this problem, we need to factor the numerator and denominator and see if anything cancels. The rules of factoring a quadratic and the special quadratic formulas can be used in this scenario.
::在此问题上, 我们需要计数数和分母, 看看是否有东西取消 。 在此情况下可以使用二次方程和特殊二次方程的计算规则 。

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In the last step, we simplified to the left hand side of the Pythagorean Identity. Therefore, this expression simplifies to 1.
::最后一步,我们简化为毕达哥里人身份的左手侧。因此,这一表达式简化为1。

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3. $\begin{array}{r}\sec \theta {\tan }^2\theta +\sec \theta \end{array}$
   ::

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First, pull out the GCF.
::首先,拿出绿色气候基金。

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: )

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Now, $\begin{array}{r}{\tan }^2\theta +1={\sec }^2\theta \end{array}$ from the Pythagorean Identities, so simplify further.
::现在,来自毕达哥利人身份的Tan2=1=sec2=2, 进一步简化。

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::seecsec2sec3

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Examples
::实例

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Example 1
::例1

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Earlier, you were asked to simplify the trigonometric function  $\begin{array}{r}\cos \theta +\cos \theta ({\tan }^2\theta )\end{array}$ .
::先前曾要求您简化三角函数 cosççççççç(tan2) 。

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Notice that the terms in the expression $\begin{array}{r}\cos \theta +\cos \theta ({\tan }^2\theta )\end{array}$ have a common factor of $\begin{array}{r}\cos \theta \end{array}$ , so start by factoring this common term out.
::请注意,Coscos(tan2)表达式中的术语有一个共同的 Cos系数,因此首先考虑这一共同术语。

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::coscos(tan2)cos(1+tan2)

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Now, use the trigonometric identity $\begin{array}{r}1+{\tan }^2\theta ={\sec }^2\theta \end{array}$ , substitute, and simplify.
::现在,使用三角特征 1+tan2sec2, 替代, 简化。

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::cos( 1+tan2) =cos( sec2) =cos( 1cos2) =1cossec

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Simplify the following trigonometric expressions.
::简化以下三角表达式。

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Example 2
::例2

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$\begin{array}{r}\cos \left(\frac{\pi }{2}-x\right)\cot x\end{array}$
::COs(%2-x)cotx

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Use the Cotangent Identity and the Cofunction Identity $\begin{array}{r}\cos \left(\frac{\pi }{2}-\theta \right)=\sin \theta \end{array}$ .
::使用余切身份和共控身份cos(2)=sin。

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$\begin{array}{r}\cos \left(\frac{\pi }{2}-x\right)\cot x\to \cancel{\sin x}\cdot \frac{\cos x}{\cancel{\sin x}}\to \cos x\end{array}$
::coms(%2-x) cotxsinxcosxsinxxxcosxxxxxxxx

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Example 3
::例3

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$\begin{array}{r}\frac{\sin \left(-x\right)\cos x}{\tan x}\end{array}$
:-x)cosxtanxxx

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Use the Negative Angle Identity and the Tangent Identity.
::使用负角身份和切换身份。

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$\begin{array}{r}\frac{\sin \left(-x\right)\cos x}{\tan x}\to \frac{-\sin x\cos x}{\frac{\sin x}{\cos x}}\to -\cancel{\sin x}\cos x\cdot \frac{\cos x}{\sin x}\to -{\cos }^{2}x\end{array}$
::xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxs2xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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Example 4
::例4

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$\begin{array}{r}\frac{\cot x\cos x}{\tan \left(-x\right)\sin \left(\frac{\pi }{2}-x\right)}\end{array}$
::xxcosxtan(- x)sin( 2- x)

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In this problem, you will use several identities.
::在这个问题上,你将使用几个身份。

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$\begin{array}{r}\frac{\cot x\cos x}{\tan \left(-x\right)\sin \left(\frac{\pi }{2}-x\right)}\to \frac{\frac{\cos x}{\sin x}\cdot \cos x}{-\frac{\sin x}{\cancel{\cos x}}\cdot \cancel{\cos x}}\to \frac{\frac{{\cos }^{2}x}{\sin x}}{-\sin x}\to \frac{{\cos }^{2}x}{\sin x}\cdot -\frac{1}{\sin x}\to -\frac{{\cos }^{2}x}{{\sin }^{2}x}\to -{\cot }^{2}x\end{array}$
::xcosxtan(xxxxxinxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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Review
::回顾

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Simplify the following expressions.
::简化下列表达式。

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1. $\begin{array}{r}\cot x\sin x\end{array}$
   ::comtxsinxxxx
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2. $\begin{array}{r}{\cos }^{2}x\tan (-x)\end{array}$
   ::COs2xtan(- x)
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3. $\begin{array}{r}\frac{\cos \left(-x\right)}{\sin \left(-x\right)}\end{array}$
   ::COs(- x) sin(- x)
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4. $\begin{array}{r}\sec x\cos (-x)-{\sin }^{2}x\end{array}$
   ::sin2 xxxxxxxxx
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5. $\begin{array}{r}\sin x(1+{\cot }^{2}x)\end{array}$
   ::exinx( 1+cot2+x)
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6. $\begin{array}{r}1-{\sin }^2\left(\frac{\pi }{2}-x\right)\end{array}$
   ::1- 辛2( 2- x)
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7. $\begin{array}{r}1-{\cos }^2\left(\frac{\pi }{2}-x\right)\end{array}$
   ::1 - COs2(%2-x)
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8. $\begin{array}{r}\frac{\tan \left(\frac{\pi }{2}-x\right)\sec x}{1-{\csc }^{2}x}\end{array}$
   ::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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9. $\begin{array}{r}\frac{{\cos }^{2}x{\tan }^{2}x-1}{{\cos }^{2}x}\end{array}$
   ::CO2 xtan2x- 1cos2x
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10. $\begin{array}{r}{\cot }^{2}x+{\sin }^{2}x+{\cos }^2(-x)\end{array}$
    ::comt2%x+sin2x+cos2(-x)
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11. $\begin{array}{r}\frac{\sec x\sin x+\cos \left(\frac{\pi }{2}-x\right)}{1+\cos x}\end{array}$
    ::=============================================================================================================================================================================================================
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12. $\begin{array}{r}\frac{\cos \left(-x\right)}{1+\sin \left(-x\right)}\end{array}$
    ::COs(- x)1+sin(- x)
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13. $\begin{array}{r}\frac{{\sin }^2\left(-x\right)}{{\tan }^{2}x}\end{array}$
    :- x)tan2 x
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14. $\begin{array}{r}\tan \left(\frac{\pi }{2}-x\right)\cot x-{\csc }^{2}x\end{array}$
    ::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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15. $\begin{array}{r}\frac{\csc x\left(1-{\cos }^{2}x\right)}{\sin x\cos x}\end{array}$
    ::cscx( 1) - cos2x) sin *xcos*xxx

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Review (Answers)
::回顾(答复)

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Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。