Step-by-step explanation:
Consider LHS
[tex] \cos(x) + \sin(x) \tan(x) = \sec(x) [/tex]
Apply quotient identies
[tex] \cos(x) + \sin(x) \times \frac{ \sin(x) }{ \cos(x) } = \sec(x) [/tex]
Multiply the fraction and sine.
[tex] \cos(x) + \frac{ \sin {}^{2} (x) }{ \cos(x) } = \sec(x) [/tex]
Make cos x a fraction with cos x as it denominator.
[tex] \cos(x) \times \cos(x) = \cos {}^{2} (x) [/tex]
so
[tex] \frac{ \cos {}^{2} (x) }{ \cos(x) } + \frac{ \sin {}^{2} (x) }{ \cos(x) } = \sec(x) [/tex]
Pythagorean Identity tells us sin squared and cos squared equals 1 so
[tex] \frac{1}{ \cos(x) } = \sec(x) [/tex]
Apply reciprocal identity.
[tex] \sec(x) = \sec(x) [/tex]
