sec(x)tan(x)=1.4142

Question

sec(x)=(1/coc(x))

Answer 1

sec(x)tan(x)=√2 (I guess),

√(1+tan²(x))tan(x)=√2.

Let y=tan(x):

y√(1+y²)=√2, squaring:

y²(1+y²)=2,

y⁴+y²-2=0=(y²+2)(y²-1).

Therefore y²=1, y=±1⇒tan(x)=±1⇒x=±π/4 (±45°).

We need to check both solutions by substituting for x in the original equation:

sec(π/4)=sec(-π/4)=√2

tan(π/4)=1⇒sec(x)tan(x)=√2, which checks out; tan(-π/4)=-1⇒sec(x)tan(x)=-√2, which doesn't check out.

However, in the first quadrant cosine and tangent are both negative so sec(π-π/4)=sec(¾π)=-√2 and tan(¾π)=-1, therefore sec(x)tan(x)=(-√2)(-1)=√2, which checks out.

Complete solution is x=¼π+2πn, ¾π+2πn, where n is any integer.

In degrees: x=45°+360n°, 135°+360n°.