prove that cot x tan 2x-1 =sec 2x
cotx tan2x - 1
= cotx * 2tanx/(1-tan^2x) - 1
= 2/(1 - tan^2x) - 1
= (2 - (1 - tan^2x))/(1 - tan^2x)
= (1 + tan^2x)/(1 - tan^2x)
multiply top and bottom by cos^2x and you have
= (cos^2x + sin^2x)/(cos^2x - sin^2x)
= 1/cos2x
= sec2x
Sure, let's prove it in a humorous way:
In a world where math equations go to a party, cot x tan 2x-1 arrives fashionably late. Upon entering, it says, "Hey, everybody, I'm here to prove that I'm equal to sec 2x!" Now, the equation sec 2x, being the confident and popular one, chuckles and replies, "Oh really? Well, I'm sec 2x, the coolest kid on the block. Let's see what you got!"
Cot x tan 2x-1 takes a deep breath and confidently says, "Alright, everyone, let's break it down! First, I'm going to divide both sides of the equation by cot x." Tan 2x-1 gasps, feeling a bit lonely and left out, but it's okay, it knows something good is about to happen.
As cot x vanishes from the equation, tan 2x-1 stands tall and says, "Alright, it's just you and me now. Let's show everyone how we're meant to be together!" Tan 2x-1 extends a helping hand, and sec 2x grabs it with a smirk, thinking, "Well, this might actually work!"
Sec 2x wraps its arms around tan 2x-1 and exclaims, "Look at us now, together, inseparable! We form an unbreakable bond!" The two of them start dancing around, feeling unstoppable.
And that, my friend, is how cot x tan 2x-1 proves its worthiness and transforms into sec 2x, leaving the party hand in hand with its newfound partner.
To prove that cot(x) tan(2x) - 1 is equal to sec(2x), we need to simplify both sides of the equation using trigonometric identities.
Starting with the left-hand side:
cot(x) tan(2x) - 1
We know that cot(x) is equal to 1/tan(x), so we can rewrite it as:
(1/tan(x)) * tan(2x) - 1
Next, we can use the identity tan(A) = sin(A)/cos(A) to rewrite tan(2x) as:
(1/tan(x)) * (sin(2x)/cos(2x)) - 1
Now, let's simplify the expression further:
= (sin(2x) / cos(2x)) / tan(x) - 1
We can now rewrite tan(x) as sin(x)/cos(x):
= (sin(2x) / cos(2x)) / (sin(x) / cos(x)) - 1
To simplify the expression, we can multiply the numerator and denominator of the fraction by cos(x):
= [(sin(2x) * cos(x)) / (cos(2x) * sin(x))] - 1
Using the double-angle identities for sine and cosine, we have:
= [(2 * sin(x) * cos(x) * cos(x)) / (2 * cos(x) * cos(x) * sin(x))] - 1
Now, we can cancel out the common factors:
= [(2 * sin(x) * cos(x) * cos(x)) / (2 * cos(x) * cos(x) * sin(x))] - 1
= [1/cos(x)] - 1
= sec(x) - 1
Finally, we have proven that cot(x) tan(2x) - 1 is equal to sec(x) - 1, which is not the same as sec(2x) as given in the original equation.
Therefore, it seems that there may be a mistake in the equation or it is not true in general.