How Can $\frac{\sec\theta}{\tan\theta}$ Be Simplified to $\csc(\theta)$?

In summary, simplifying a trig equation involves using trigonometric identities and basic algebraic rules to manipulate the equation into a simpler form. This can include using the Pythagorean identities, double angle formulas, and trigonometric ratios to rewrite the equation in a more simplified manner. A trig equation is considered simplified when it cannot be further reduced or rewritten using trigonometric identities or basic algebraic rules. This means that there are no more terms that can be combined or simplified, and the equation is in its most simplified form. It is possible to simplify a trig equation with variables, but it may be necessary to solve for the variables first. While there is no set formula for simplifying a trig equation, it is helpful to follow a systematic
  • #1
tmt1
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I have $\frac{sec\theta}{tan\theta}$. How can I simplify it to get $\csc\left({\theta}\right)$ ?
 
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  • #2
tmt said:
I have $\frac{sec\theta}{tan\theta}$. How can I simplify it to get $\csc\left({\theta}\right)$ ?
I prefer to change everything to sines and cosines.
\(\displaystyle \frac{sec( \theta )}{tan( \theta )} = \frac{ \frac{1}{cos( \theta )}}{ \frac{sin( \theta )}{cos( \theta )}}\)

\(\displaystyle = \frac{ \frac{1}{cos( \theta )}}{ \frac{sin( \theta )}{cos( \theta )}} \cdot \frac{ cos( \theta )}{cos( \theta )}\)

\(\displaystyle = \frac{1}{sin( \theta )} = csc( \theta )\)

With, of course, the restrictions that \(\displaystyle sin( \theta ) \neq 0\) and \(\displaystyle cos( \theta ) \neq 0\)

-Dan
 

FAQ: How Can $\frac{\sec\theta}{\tan\theta}$ Be Simplified to $\csc(\theta)$?

1. What is the process for simplifying a trig equation?

Simplifying a trig equation involves using trigonometric identities and basic algebraic rules to manipulate the equation into a simpler form. This can include using the Pythagorean identities, double angle formulas, and trigonometric ratios to rewrite the equation in a more simplified manner.

2. How do you know when a trig equation is simplified?

A trig equation is considered simplified when it cannot be further reduced or rewritten using trigonometric identities or basic algebraic rules. This means that there are no more terms that can be combined or simplified, and the equation is in its most simplified form.

3. Can you simplify a trig equation with variables?

Yes, it is possible to simplify a trig equation with variables. The same principles of using trigonometric identities and basic algebraic rules apply. However, it may be necessary to solve for the variables first before simplifying the equation.

4. Are there any specific steps to follow when simplifying a trig equation?

While there is no set formula for simplifying a trig equation, it is helpful to follow a systematic approach. This may include identifying any known trigonometric identities that can be used, combining like terms, and grouping terms together to make the equation more manageable.

5. Why is it important to simplify a trig equation?

Simplifying a trig equation can make it easier to solve and understand. It can also help to reveal patterns and relationships between different trigonometric functions. In addition, simplified trig equations are often used in real-world applications such as physics, engineering, and astronomy.

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