Simplify the following expression as far as
possible:

Ask by Ortega Bernard. in South Africa

Nov 11,2024

Question

Simplify the following expression as far as possible: $$ \frac{1}{\sin x}-\frac{\cos x}{	an x} $$

UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ 1/\sin(x)-\cos(x)/	an(x) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{1}{\sin\left(xight)}-\frac{\cos\left(xight)}{	an\left(xight)}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{	an\left(xight)}{\sin\left(xight)	an\left(xight)}-\frac{\cos\left(xight)\sin\left(xight)}{	an\left(xight)\sin\left(xight)}$$
- step2: Rewrite the expression:
  $$\frac{	an\left(xight)}{\sin\left(xight)	an\left(xight)}-\frac{\cos\left(xight)\sin\left(xight)}{\sin\left(xight)	an\left(xight)}$$
- step3: Transform the expression:
  $$\frac{	an\left(xight)-\cos\left(xight)\sin\left(xight)}{\sin\left(xight)	an\left(xight)}$$
- step4: Transform the expression:
  $$\frac{\frac{\sin\left(xight)-\cos^{2}\left(xight)\sin\left(xight)}{\cos\left(xight)}}{\sin\left(xight)	an\left(xight)}$$
- step5: Transform the expression:
  $$\frac{\frac{\sin\left(xight)-\cos^{2}\left(xight)\sin\left(xight)}{\cos\left(xight)}}{\frac{\sin^{2}\left(xight)}{\cos\left(xight)}}$$
- step6: Multiply by the reciprocal:
  $$\frac{\sin\left(xight)-\cos^{2}\left(xight)\sin\left(xight)}{\cos\left(xight)}	imes \frac{\cos\left(xight)}{\sin^{2}\left(xight)}$$
- step7: Rewrite the expression:
  $$\frac{\sin\left(xight)\left(1-\cos^{2}\left(xight)ight)}{\cos\left(xight)}	imes \frac{\cos\left(xight)}{\sin^{2}\left(xight)}$$
- step8: Reduce the fraction:
  $$\left(1-\cos^{2}\left(xight)ight)	imes \frac{1}{\sin\left(xight)}$$
- step9: Multiply the terms:
  $$\frac{1-\cos^{2}\left(xight)}{\sin\left(xight)}$$
- step10: Transform the expression:
  $$\frac{\sin^{2}\left(xight)}{\sin\left(xight)}$$
- step11: Divide the numbers:
  $$\sin^{2-1}\left(xight)$$
- step12: Calculate:
  $$\sin\left(xight)$$
The simplified form of the expression $$ \frac{1}{\sin x}-\frac{\cos x}{	an x} $$ is $$ \sin x $$.
$$ \sin x $$

Simplify the following expression as far as
possible:

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