Solve in terms of sine and cosine:
sec(x) csc(x)- sec(x) sin(x)
so far I have:
1/cos(x) 1/sin(x) - 1/cos(x) sin(x)
I am not sure where to go to from there. The book says the answer is cot(x) or cos(x)/sin(x)
Thank you in advance.
sec(x) csc(x)- sec(x) sin(x)
= (1/cos x)(1/sinx) - sinx/cosx
= [(1/sinx)-sinx]/cos x
= [1 - sin^2x]/(sin x cos x)
= cos^2 x/(sin x cos x)
= cosx/sinx = cot x
Dang, I see where I messed up. Thank you so much.
whoops, sorry everyone. The original message was supposed to be a reply to another post.
To solve the expression in terms of sine and cosine, we can simplify and manipulate the equation step by step.
Starting with:
sec(x) csc(x) - sec(x) sin(x)
We can rewrite sec(x) as 1/cos(x) and csc(x) as 1/sin(x):
(1/cos(x)) * (1/sin(x)) - (1/cos(x)) * sin(x)
Now we can combine the fractions under a common denominator:
(1 * sin(x) - (1/cos(x)) * sin(x)) / (cos(x) * sin(x))
Simplifying further, we can factor out sin(x) from the numerator:
(sin(x) - (1/cos(x)) * sin(x)) / (cos(x) * sin(x))
Now let's rewrite (1/cos(x)) as sec(x) and substitute it back into the equation:
(sin(x) - sec(x) * sin(x)) / (cos(x) * sin(x))
Next, we can factor out sin(x) from the numerator:
sin(x) * (1 - sec(x)) / (cos(x) * sin(x))
Notice that sin(x) cancels out:
(1 - sec(x)) / cos(x)
Since sec(x) is defined as 1/cos(x), we can substitute it back in:
(1 - (1/cos(x))) / cos(x)
Now, simplify by combining and subtracting the fractions:
(1 - 1/cos(x)) / cos(x)
Multiplying the numerator and denominator by cos(x):
(cos(x) - 1) / cos(x)^2
Finally, we can divide both terms by cos(x) to get the final expression:
(cos(x) - 1) / cos(x)^2 = cot(x)
So, the solution to the equation in terms of sine and cosine is cot(x) or cos(x)/sin(x).