Add or subtract as indicated, and then simplify as much as possible. Leave your answer in terms of
sines and cosines only,

Ask by Mcfarlane Ayala. in the United States

Jan 24,2025

Question

Add or subtract as indicated, and then simplify as much as possible. Leave your answer in terms of sines and cosines only, $$ \frac{\cos (x)}{\sin x}+\frac{\sin (x)}{\cos (x)}=\square $$

UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \frac{\cos (x)}{\sin x}+\frac{\sin (x)}{\cos (x)} $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\cos\left(xight)}{\sin\left(xight)}+\frac{\sin\left(xight)}{\cos\left(xight)}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{\cos\left(xight)\cos\left(xight)}{\sin\left(xight)\cos\left(xight)}+\frac{\sin\left(xight)\sin\left(xight)}{\cos\left(xight)\sin\left(xight)}$$
- step2: Rewrite the expression:
  $$\frac{\cos\left(xight)\cos\left(xight)}{\sin\left(xight)\cos\left(xight)}+\frac{\sin\left(xight)\sin\left(xight)}{\sin\left(xight)\cos\left(xight)}$$
- step3: Transform the expression:
  $$\frac{\cos\left(xight)\cos\left(xight)+\sin\left(xight)\sin\left(xight)}{\sin\left(xight)\cos\left(xight)}$$
- step4: Multiply the terms:
  $$\frac{\cos^{2}\left(xight)+\sin\left(xight)\sin\left(xight)}{\sin\left(xight)\cos\left(xight)}$$
- step5: Multiply the terms:
  $$\frac{\cos^{2}\left(xight)+\sin^{2}\left(xight)}{\sin\left(xight)\cos\left(xight)}$$
- step6: Transform the expression:
  $$\frac{1}{\sin\left(xight)\cos\left(xight)}$$
- step7: Transform the expression:
  $$\frac{1}{\frac{\sin\left(2xight)}{2}}$$
- step8: Multiply by the reciprocal:
  $$1	imes \frac{2}{\sin\left(2xight)}$$
- step9: Multiply:
  $$\frac{2}{\sin\left(2xight)}$$
- step10: Transform the expression:
  $$2\csc\left(2xight)$$
The simplified expression is $$2\csc(2x)$$.
$$ \frac{\cos (x)}{\sin x}+\frac{\sin (x)}{\cos (x)} = 2\csc(2x) $$

Add or subtract as indicated, and then simplify as much as possible. Leave your answer in terms of
sines and cosines only,

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