Question

If sec θ + tan θ = m, then the value of sec⁴ θ – tan⁴ θ – 2 sec θ tan θ is ______.

Answer 1

If sec θ + tan θ = m, then the value of sec⁴ θ – tan⁴ θ – 2 sec θ tan θ is `underlinebb(1/m^2)`.

Explanation:

sec⁴θ – tan⁴θ = (sec²θ – tan²θ)(sec²θ + tan²θ) 

= 1·(sec²θ + tan²θ)   ...(Since sec²θ – tan²θ = 1)

Using this, sec⁴θ – tan⁴θ – 2 secθ tanθ 

= sec²θ + tan²θ – 2 secθ tanθ 

= (secθ – tanθ)² 

But (secθ + tanθ)(secθ – tanθ) = 1.

So `secθ - tanθ = 1/(secθ + tanθ) = 1/m`. 

Therefore the expression equals `(1/m)^2 = 1/m^2`.