Trig identity that I'm missing

[member 392791]

I am trying to integrate -tan(x)sec^2(x) and getting -tan^2(x) / 2. When I put it in wolfram alpha it gets the same answer when I press show solution, but without pressing it it shows -sec(x)/2.

So I am wondering, is it the case tan^2(x) = sec(x)?? I don't remember this as a correct trig identity

http://www.wolframalpha.com/input/?i=-tan%28x%29sec^2%28x%29+integral

Notice the difference in answer when you press for the show step by step solution

 

[MarneMath]

I'm reading (1/2)[-sec(x)]^2 + C, in which case they might've used [tan(x)]^2 = [sec(x)]^2-1

 

[member 392791]

Ahh I was misreading it, thanks.

 

[Mark44]

In indefinite integral can have different answers, depending on the technique that is used, but the answers can differ by at most a constant.

When you integrated -tan(x)sec²(x) you got -(1/2)tan²(x). Wolfram's answer was -(1/2)sec²(x). Since sec²(x) = tan²(x) + 1, your answer and Wolfram's answer differ by a constant.

 

[CellCoree]

Hello,

It appears that the discrepancy in the answer is due to the use of different trigonometric identities. The identity -tan^2(x)/2 is equivalent to -sec(x)/2, as tan^2(x) is equal to sec^2(x) - 1. This is a commonly used identity in trigonometric integration.

However, when you press "show step by step solution" on Wolfram Alpha, it uses the identity -tan(x)sec(x) = -1, which simplifies the integral to -1/2. Both answers are correct, but the second one is considered a more simplified form.

To answer your question, tan^2(x) is not equal to sec(x). The correct identity is tan^2(x) = sec^2(x) - 1.

I hope this clarifies any confusion. Keep up the good work in your integration practice!