Question

Simplify the left side of equation so it looks like the right side. cos(x) + sin(x) tan(x) = sec (x)

Answer 1

Explanation:

Consider LHS

`\cos(x) + \sin(x) \tan(x) = \sec(x)`

Apply quotient identies

`\cos(x) + \sin(x) * ( \sin(x) )/( \cos(x) ) = \sec(x)`

Multiply the fraction and sine.

`\cos(x) + \frac{ \sin {}^(2) (x) }{ \cos(x) } = \sec(x)`

Make cos x a fraction with cos x as it denominator.

`\cos(x) * \cos(x) = \cos {}^(2) (x)`

so

`\frac{ \cos {}^(2) (x) }{ \cos(x) } + \frac{ \sin {}^(2) (x) }{ \cos(x) } = \sec(x)`

Pythagorean Identity tells us sin squared and cos squared equals 1 so

`(1)/( \cos(x) ) = \sec(x)`

Apply reciprocal identity.

`\sec(x) = \sec(x)`