Question

If sec theta + tan theta= m, then the value of sec^4 theta -tan^4 theta -2sec theta tan theta is:
a. m
b. m
c. 1/m
d. None of these​

Answer 1

$option : (d)$

Explanation:

$Given, \: \: sec \theta + tan \theta = m \\ we \: \: know \: \: that \: \: \: \boxed{sec^{2} \theta - tan^{2} \theta = 1 } \\ or, \: \: \: (sec \theta + tan \theta )(sec \theta - tan \theta ) = 1 \\or, \: \: \: m(sec \theta - tan \theta ) = 1 \\ or, \: \: \: sec \theta - tan \theta = \frac{1}{m} \\ Now, \: \: \: \: sec^4 \theta -tan^4 \theta -2sec \theta tan \theta \\ = \: (sec^{2} \theta + tan^{2} \theta )(sec^{2} \theta - tan^{2} \theta ) -2sec \theta tan \theta \\ = \: (sec^{2} \theta + tan^{2} \theta ) \times 1 -2sec \theta tan \theta \\ = \: sec^{2} \theta + tan^{2} \theta -2sec \theta tan \theta \\ = (sec \theta - tan \theta)^{2} \\ putting \: \: value \\ = ( \frac{1}{m} )^{2} = \frac{1}{ {m}^{2} }$