Friday, January 5, 2001

How do I solve (sec^2(x))+(csc^2(x)) ?

I am really stuck on getting a common denominator. I get to 1/cos^2x+1/sin^2x , but than I can't figure out how to multiply the terms together to get a common denominator. I keep getting (cos^2x)(sin^2x)/(cos^2x)(sin^2x) , because I know I have to do the same to the top as I do to the bottom.
If someone could explain why I am doing wrong and solve the problem I would greatly appreciate it !
(I know that I am solving this wrong because it would simplify to 1, and that is not an option in my practice booklet.)

Answer on How do I solve (sec^2(x))+(csc^2(x)) ?

sec?(x) + csc?(x)

Convert into sin/cos:
1/cos?(x) + 1/sin?(x)

Multiply the first term by sin?(x)/sin?(x):
sin?(x)/[sin?(x)cos?(x)] + 1/sin?(x)

Multiply the second term by cos?(x)/cos?(x):
sin?(x)/[sin?(x)cos?(x)] + cos?(x)/[sin?(x)cos?(x)]

Add both terms, since they have a common denominator:
[sin?(x) + cos?(x)] / [sin?(x)cos?(x)]

The numerator can be reduced to 1:
1/[sin?(x)cos?(x)]

This can be rewritten as:
csc?(x)sec?(x)

Or as:
2/sin?(2x) = 2csc?(2x)