Hi again all,
sec x tan x - cos x cot x = sin x
I've been looking at this for about 10 mins trying to figure out what first steps to take. Is there a general rule I should follow in terms of what to do first?
I know the trig identities and I went about trying to apply some but I have trouble knowing which to use and how to continue, I will write down what I have in my book that I stopped at:
(sin x/cos^2x) - cos x (1/tanx) = sin x
from there I tried to apply pythagorean theorem for cos^2x but I just dont have enough experience to work it out further, I did keep trying and I will keep trying but for now I am reviewing my whole term in preparation for final. I really don't mean to ask for answers, I DO want to understand these things and I've done 50 exercises just today on it.
how about looking at the left hand side of the equation:
since $\displaystyle cotx=\frac{cosx}{sinx}$, you can write
$\displaystyle \frac{sinx}{cos^{2}x} - \frac{cos^{2}x}{sinx} = ....$
can you move ahead now? You should be able to get to the destination. Reply back if you still have problems
I seem to have gotten as far as:
(sin^2 x - (1-sin^2 x)^2)/((1-sin^2 x)(sin x)) = sin x
I got that from using the cos^2 x pythagorean identity and then trying to treat them like regular subtraction for fractions (use common denominator).
This is the last section of the precal course and I will admit I have trouble with trig identities
$\displaystyle \frac{(sin^2 x - (1-sin^2 x)^2)}{((1-sin^2 x)(sin x))} = sinx$
$\displaystyle (sin^2 x - (1-2sin^2 x+sin^4x) = sinx (1-sin^2 x) (sinx)$
$\displaystyle sin^2 x - 1 + 2sin^2 x- sin^4x = sin^2x - sin^4x$
cancel the similar terms on the left side and the right side, then you can try solving for x now. Reply if you have any errors.
[0, 2pi). thats what you said the interval of x was! And the answers you have gotten do not look correct.
$\displaystyle sin^2 x - 1 + 2sin^2 x- sin^4x = sin^2x - sin^4x$
this gives:
$\displaystyle -1 + 2sin^2 x = 0$
$\displaystyle 2sin^2x = 1$
$\displaystyle sin^2x = \frac{1}{2}$
now find out what sinx would be? Then try to figure out the values of x from $\displaystyle [0, 2\pi)$ for which your result is true!