How do I solve this? My work has led me to a dead end.
tan(45-x) + cot(45-x) =4
my work:
(tan45 - tanx)/(1+ tan45tanx) + (cot45 - cotx)/(1 + cot45cotx) = 4
(1-tanx)/(1+tanx) + (1-cotx)/(1+cotx) = 4
Then I found a common denominator, giving me this:
(2-2cotxtanx)/(1+cotx+tanx+cotxtanx) =4
I can't cancel the cotxtanx's because they're in the middle of a term, right? So where do I go from here? Or did I make a mistake somewhere?
This is how I would soleve it. First put 45 - x = y. Then:
Sin(y)/Cos(y) + Cos(y)/Sin(y) = 4 -->
[Sin^2(y) + Cos^2(y)]/(Sin(y)Cos(y)) = 4
1/(Sin(y)Cos(y)) = 4 --->
Sin(2y) = 1/2
To solve the equation tan(45-x) + cot(45-x) = 4, you made some progress with your work. However, there is a small mistake when simplifying the expression. Let's take a look at it again:
Start with your work:
(tan45 - tanx)/(1 + tan45tanx) + (cot45 - cotx)/(1 + cot45cotx) = 4
Now, let's simplify it further:
[(1 - tanx)/(1 + tanx)] + [(1 - cotx)/(1 + cotx)] = 4
To find a common denominator for the fractions, multiply the individual denominators:
[(1 - tanx)(1 + cotx)] + [(1 - cotx)(1 + tanx)] = 4(1 + tanx)(1 + cotx)
Expanding the parentheses:
(1 + cotx - tanx - tanxcotx) + (1 + tanx - cotx - cotxtanx) = 4(1 + tanx + cotx + cotxtanx)
Combining like terms:
2 - 2tanxcotx = 4 + 4tanx + 4cotx + 4cotxtanx
Now, let's simplify it further:
2 - 2cotx(tanx) = 4 + 4(tanx + cotx) + 4cotxtanx
Now, you mentioned that you can't cancel the cotxtanx because they are in the middle of a term. That's correct. We can't cancel them out at this stage.
To continue solving this equation, we can simplify it further by gathering the terms with cotx and tanx on one side and the constant terms on the other side:
2 - 4 - 4(tanx + cotx) = 2cotx(tanx) + 4cotxtanx
Combining like terms:
-2 - 4(tanx + cotx) = 2cotx(tanx) + 4cotxtanx
Now, let's factor out cotx(tanx):
-2 - 4(tanx + cotx) = 2cotx(tanx + 2tanx)
Combining like terms:
-2 - 4(tanx + cotx) = 2cotx(3tanx)
Now, we have an equation in terms of only cotx, tanx, and constants. To solve it further, you can substitute any value for tanx and then solve for cotx, or vice versa. Alternatively, you can plot the graph of both sides of the equation and find the intersection points.