How can you use the identity 1+tan^2x = sec^2x to simplify the equation?

[Trizz]

Homework Statement

sec^2(x) tan^2(x) + sec^2(x) = sec^4(x)

Homework Equations

sin^2 + cos^2 = 1
1+tan^2 = sec^2
1+cot^2 = csc^2

The Attempt at a Solution

First, I changed everything to sin and cos to try and make it clearer.

1/cos^2 * sin^2/cos^2 + 1/cos^2 = sec^4
sin^2/cos^4 + 1/cos^2

Then I multiplied by the common denominator, cos^2

sin^2 * cos^2/cos^6 + cos^2/cos^4

Where do I go from here??

 

[Bohrok]

You don't need to multiply the first term by cos²x/cos², just the second so they have a common denominator. Then add the fractions and use an identity.

 

[Mentallic]

You don't need to change everything into sinx and cosx to makes things clearer. Even if you can't think about what secx is without thinking of 1/cosx you can still solve this problem:

Set the equation to 0, then factorize by a common factor which should obviously be $sec^2x$. Now use an identity.
At this point, if you're confused about what has happened just think about this: for the equation x-x=0, you can have any value of x to satisfy the equation. This means all x values are the equation's solutions.

But also remember that $0/0\neq 0$ so make sure to show that the solutions to $cosx= 0$ are excluded from the solutions in the original equation (This is from the first factor $sec^2x$).