Question

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

Answer 1

$$\begin{gathered} {\sec }^4\theta -{\tan }^4\theta -2\sec \theta \tan \theta \\ =\left({\sec }^2\theta -{\tan }^2\theta \right)\left({\sec }^2\theta +{\tan }^2\theta \right)-2\sec \theta \tan \theta \\ ={\sec }^2\theta +{\tan }^2\theta -2\sec \theta \tan \theta \left(1+{\tan }^2\theta ={\sec }^2\theta \right) \\ ={\left(\sec \theta -\tan \theta \right)}^2 \end{gathered}$$
$$\begin{gathered} ={\left[\frac{\left(\sec \theta +\tan \theta \right)\left(\sec \theta -\tan \theta \right)}{\sec \theta +\tan \theta }\right]}^2 \\ ={\left(\frac{{\sec }^2\theta -{\tan }^2\theta }{\sec \theta +\tan \theta }\right)}^2 \\ ={\left(\frac{1}{m}\right)}^2 \\ =\frac{1}{m^2} \end{gathered}$$

If sec θ + tan θ = m, then the value of sec⁴ θ – tan⁴ θ – 2 sec θ tan θ is $\underline{\frac{1}{m^2}}$.