Question

Find the solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.)sec θ − tan θ = cos θ

Answer 1

To solve sec θ − tan θ = cos θ within the interval [0, 2π), we substitute trigonometric identities and simplify to find that θ equals 0 and π.

Step-by-step explanation:

The original equation is sec θ − tan θ = cos θ. We can rewrite sec θ as 1/cos θ and tan θ as sin θ / cos θ to get:

1/cos θ − (sin θ / cos θ) = cos θ

Multiplying through by cos θ to clear the fractions, we have:

1 − sin θ = cos² θ

But cos² θ is equal to 1 − sin² θ, hence:

1 − sin θ = 1 − sin² θ

Subtracting 1 from both sides and simplifying, we get:

sin² θ − sin θ = 0

Factoring out sin θ:

sin θ(sin θ − 1) = 0

So either sin θ = 0 or sin θ − 1 = 0. Hence, θ = 0 or θ = π. Therefore, the solutions in the interval [0, 2π) are 0 and π.

Answer 2

π/2

Step-by-step explanation:

Given the expression;

secθ − tan θ = cosθ

We are to find the solution in the interval [0, 2π).This is as shown;

From trigonometry identity;

secθ = 1/cosθ

tanθ = sinθ/cosθ

Substitute into the formula;

secθ − tan θ = cosθ

1/cosθ-sinθ/cosθ = cosθ

Multiply through by cosθ

1 - sinθ = cos²θ

1-sinθ = (1-sin²θ)

1-sin²θ-1+sinθ =0

-sin²θ+sinθ = 0

sin²θ = sinθ

sinθ = 1

θ = arcsin 1

θ = 90

θ = π/2

Hence the solution is π/2