Help with Life of Fred trig? Page 111

[amys]

My son can't figure out how/why a step in the problem at the bottom of page 111 in Life of Fred trigonometry was arrived at. I'm no help, either. Can anyone help? 

 

To show that 1 + cot² x = csc² x

 

Changing to sines and cosines (ok here)              1 + cos²x/sin²x = 1/sin²x

Adding the fractions (this is what he doesn't get)   (sin²x + cos²x)/sin²x

Since sin²x + cos²x = 1 (ok here too)                     1/sin²x

 

He doesn't understand how the term in the second step is derived.  Anyone?

Thanks!
Amy

[regentrude]

 

My son can't figure out how/why a step in the problem at the bottom of page 111 in Life of Fred trigonometry was arrived at. I'm no help, either. Can anyone help? 

 

To show that 1 + cot² x = csc² x

 

Changing to sines and cosines (ok here)              1 + cos²x/sin²x = 1/sin²x

Adding the fractions (this is what he doesn't get)   (sin²x + cos²x)/sin²x

Since sin²x + cos²x = 1 (ok here too)                     1/sin²x

 

He doesn't understand how the term in the second step is derived.  Anyone?

Thanks!

Amy

 

Two fractions with different denominators must be brought on the common denominator to be added.

1 + cos²x/sin²x

 

we need to make the 1 into a fraction with the denominator sin²x. So, we multiply top and bottom of 1/1 with sin²x

and get sin²x/sin²x (which is still equal to 1)

 

Now we can add:

sin²x/sin²x+cos²x/sin²x = add numerators of fractions with like denominators: (sin²x+cos²x)/sin²x

 

[amys]

Thank you so much!!