Solve sec2x equals secx plus 2?

Answer 1

You must mean, either,

(1) sec2 x = sec x + 2, or

(2) sec(2x) = sec x + 2.

Let's first assume that you mean:

sec2 x = sec x + 2;whence,

if we let s = sec x, we have,

s2 = s + 2,

s2 - s - 2 = (s - 2)(s + 1) = 0, and

s = 2 or -1; that is,

sec x = 2 or -1.

As, by definition,

cos x = 1/sec x ,

this means that

cos x = ½ or -1.

Therefore, providing that the first assumption is correct,

x = 60°, 180°, or 300°; or,

if you prefer,

x = ⅓ π, π, or 1⅔ π.

Now, let's assume, instead, that you mean:

sec(2x) = sec x + 2;whence,

1/(cos(2x) = (1/cos x) + 2.

If we let c = cos x,

then we have the standard identity,

2c2 - 1 = cos (2x); and,

thus, it follows that

1/(cos(2x) = 1/(2c2 - 1)

= (1/c) + 2 = (1 + 2c)/c.

This gives,

1/(2c2 - 1) = (1 + 2c)/c;

(2c2 - 1)(2c + 1) = 4c3 + 2c2 - 2c - 1 = c; and

4c3 + 2c2 - 3c - 1 = (c + 1)(4c2 - 2c - 1) = 0.

As our concern is only with real roots,

c = cos x = -1; and,

therefore, providing that the second assumption is correct,

x = 180°; or,

if you like,

x = π.