-tan2 x + sec2 x = 1

Question

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation. 

Answer 1

If you have trouble remembering Pythagorean Identities, you don't need to :)

You can get them all from the main trig identity sin² x +cos² x = 1

 

First, divide both sides of this identity by sin² x. You get:

sin² x/sin² x +cos² x/sin² x = 1/sin² x

1 + cot² x = csc² x

 

Recognize this one? This is another variation of the Pythagorean Identity.

 

Second, divide both sides of the first identity by cos² x. You get:

sin² x/cos² x +cos² x/cos² x = 1/cos² x
tan² x + 1 = sec² x

 

This is the third Pythagorean Identity.

 

Then, it's a matter of pattern matching. You can solve any identity for what you need in the expression you are given and substitute. The solution above shows this substitution for sec²x.

 

I don't like memorizing stuff I don't understand. So, I always share with my students ways to understand the stuff so that they can always "get it back" if they forget. If I could insert a picture here, I could show you where the first identity comes from!

 

 

Answer 2

I assume you meant:  -tan² x + sec² x = 1

 

You can use one the "Pythagorean identities" to prove this:  tan² x + 1 = sec² x

 

So you would do the following:

 

-tan² x + sec² x = 1

-tan² x + tan² x + 1 = 1

 

Then see that -tan² x and  tan² x cancel each other, so that leaves you with:

 

1 = 1

 

Since that is true, the original identity has been proven