Solve Math Problem: Simplify \sin\theta\sec\theta+\cos\theta\csc\theta

[PanTh3R]

Homework Statement

simplify:
$$\sin\theta\sec\theta+\cos\theta\csc\theta$$

Homework Equations

Reciprocal identities, Quotient identities, Pythagorean identities

The Attempt at a Solution

$$\sin\theta\sec\theta+\frac{1}{\sec\theta}\frac{1}{\sin\theta}$$

$$\sin\theta\sec\theta+\frac{1}{\sin\theta\sec\theta}$$

and this is where i get stuck...can i get help from anyone?

 

[tiny-tim]

Hi PanTh3R!

(have a theta: θ )

You need to be systematic …

keep all the sin and cos, get rid of the sec and csc.

Try again. ​

 

[PanTh3R]

so i get...

$$\sin\theta\frac{1}{\cos\theta}+cos\theta\frac{1}{\sin\theta}$$

then i put each in one fraction right?

$$\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}$$

to

$$\tan\theta+cot\theta$$

is that the simplest it can get?

 

[tiny-tim]

Hi PanTh3R!

This is elementary algebra …

put both fractions over the same denominator (the LCM). ​

 

[PanTh3R]

im sorry I am really bad with these i just started doing them um so i do

$$\frac{\sin^2\theta+\cos\theta}{\cos\theta\sin\theta}$$

i still don't get it won't that make it more complicated?

 

[tiny-tim]

Bottom right, top wrong.

 

[PanTh3R]

would it be...

$$\frac{\sin^2\theta+\cos^2\theta}{\cos\theta\sin\theta}$$

then

$$\frac{1-\cos^2+\cos^2\theta}{\cos\theta\sin\theta}$$

into

$$\frac{1}{\cos\theta\sin\theta}$$

i feel so frustrated sry

 

[Bohrok]

Looks like that's the simplest you can get it.

 

[icystrike]

Looks like that's the simplest you can get it.

$$2cosec(2\theta)$$ would seems better.

 

[Mentallic]

It depends if the OP has been exposed to double-angles yet. Trigonometric simplifications of this form are commonly taught before the student ever learns that $2sin\theta cos\theta=sin(2\theta)$