Hello, I need help with solving these trig identities. All help is appreciated!
1. Establish the Identity:
(cos(2x)/1+sin(2x)) = ((cotx-1)/(cot+1))
2. Establish the Identity
((sec²x-tan²x+tanx)/(secx)) = (sinx+cosx)
basic method:
test to see if it is actually true by picking a non-standard angle. That is, don't use something like 45° or 90°
usually I start with the more complicated side and try to reduce it so it becomes the other side.
#1.
LS = (cos(2x))/(1+sin(2x)) , I notice that the RS has x as the angle and not 2x, so use the formulas that reduces trig angles form 2x to x
I also corrected the brackets in your LS
= (cos^2 x - sin^2 x)/(sin^2 x + cos^2 x + 2sinxcosx), I replaced 1 with sin^2x + cos^2x
= (cosx + sinx)(cosx - sinx) / (cosx + sinx)^2
= (cosx - sinx) / (cosx + sinx)
now I have to get those cotangents in there, remember cotx = cosx/sinx
so let's divide top and bottom by sinx, realizing that sinx/sinx = 1
= (cosx/sinx - sinx/sinx) / (cosx/sinx + sinx/sinx)
= (cotx - 1) / (cotx + 1)
= RS
#2, for this one, hint: sec^2 x = 1 + tan^2 x
Thank you so much Reiny! That was a lot of help :)
Sure, I can help you with solving these trigonometric identities. Let's start with the first one:
1. To establish the identity:
(cos(2x)/(1+sin(2x))) = ((cot(x)-1)/(cot(x)+1))
We'll work on one side of the equation at a time.
On the left side of the equation, we have cos(2x)/(1+sin(2x)). We can rewrite cos(2x) as cos²(x) - sin²(x) by using the double-angle formula for cosine. So, the left side becomes:
((cos²(x) - sin²(x))/(1+sin(2x)))
Next, let's factor out a minus sign from the denominator to simplify things:
((cos²(x) - sin²(x))/(-sin(2x) - 1))
Now, let's use the double-angle formula for sine, which states that sin(2x) = 2sin(x)cos(x). By substituting this into the expression, we get:
((cos²(x) - sin²(x))/(-2sin(x)cos(x) - 1))
Now, let's simplify the numerator by using the difference of squares formula, which states that a² - b² = (a + b)(a - b):
((cos(x) - sin(x))(cos(x) + sin(x))/(-2sin(x)cos(x) - 1))
We can see that we have (cos(x) + sin(x)) in both numerator and denominator. We can cancel these out:
(-(cos(x) - sin(x))/-1)
Finally, we can multiply both numerator and denominator by -1 to get:
(cos(x) - sin(x))/1
This matches the right side of the equation ((cot(x) - 1)/(cot(x) + 1)), so the identity is established.
Now, let's move on to the second identity:
2. To establish the identity:
((sec²(x) - tan²(x) + tan(x))/sec(x)) = sin(x) + cos(x)
We'll once again work on one side of the equation at a time.
On the left side of the equation, we have ((sec²(x) - tan²(x) + tan(x))/sec(x)). First, let's rewrite sec²(x) as 1 + tan²(x) by using the Pythagorean identity for secant. So, the left side becomes:
((1 + tan²(x) - tan²(x) + tan(x))/sec(x))
We can simplify the expression by canceling out the tan²(x) terms:
((1 + tan(x))/sec(x))
Next, let's rewrite sec(x) as 1/cos(x):
((1 + tan(x))/(1/cos(x)))
To simplify the expression further, let's multiply the numerator and denominator by cos(x) to get rid of the division:
((1 + tan(x))(cos(x))/1)
Now, let's simplify the numerator by using the distributive property:
(cos(x) + sin(x)cos(x))/1
Lastly, we can rewrite sin(x)cos(x) as 1/2sin(2x) using the double-angle formula for sine:
(cos(x) + 1/2sin(2x))
This matches the right side of the equation (sin(x) + cos(x)), so the identity is established.
I hope this helps! If you have any further questions or need clarification, feel free to ask.