Simplify 1-tan^4(theta)+sec^4(theta)?

Question

I am very confused as to what identity I need to use to simplify this equation.

Answer 1

1- tan⁴θ + sec⁴θ 

 

= 1 -( sin⁴ θ/ cos ⁴θ ) + (1/ cos ⁴θ)                 as tan θ  =  Sin θ / Cosθ and Sec θ = 1/Cos θ

 

 

=( Cos⁴θ -Sin⁴θ +1 ) / Cos⁴ θ        as Cos⁴ θ - Sin ⁴θ  = (Cos ² θ -Sin ²θ  ) ( Cos ² θ + Sin ² θ) =(Cos ² θ -Sin ²θ )

 

= ( Cos²θ - Sin² θ +1)/( Cos ⁴ θ  )

 

 

= 2 Cos² θ/ Cos ⁴θ = 2/Cos²θ =2 Sec² θ

 

Answer 2

Let  x=θ

 

 

1 - tan⁴x + sec⁴x

 

Using the identity          1 + tan²x = sec²x

 

                                         tan²x = sec²x - 1    

 

 

 

1 - (sec² - 1)² + sec⁴x =

 

1 - (sec⁴x - 2sec²x + 1) + sec⁴x =

 

2sec²x

Answer 3

For this problem I would use sec²θ = 1 + tan²θ. The problem simplifies rather nicely in a few steps. If you need more help, let us know.

 

In the future, to get you started, look for trig functions in the problem and see if there is a trig identity that has most (or all, as in this case) of it. For this problem in particular, tangent and secant raised to powers is a dead giveaway. After working problems like this for a while you will start looking for things like this.

Answer 4

sec⁴ = (1 + tan²)²

sec⁴ = 1 + 2tan² + tan⁴

1 - tan⁴ + 1 + 2tan² + tan⁴

2 + 2tan²

2(1 + tan²)

2sec²