Question

Rewrite each expression in terms of sine and cosine only.

Answer 1

Answer:

1. Cosθ / SineθCosθ

2. Sineθ / Cos²θSineθ

3. Cosθ / Sineθ

4. Cos²θ + Sin²θ – Sin²θ

Step-by-step explanation:

1. SecθCotθ

Recall

Sec θ = 1/Cos θ

Cot θ = 1/Tan θ

But Tan θ = Sine θ / Cos θ

Thus,

Cot θ = 1 ÷ Sine θ / Cos θ

Cot θ = 1 × Cos θ / Sine θ

Cot θ = Cos θ / Sine θ

Therefore,

SecθCotθ = 1/Cos θ × Cos θ / Sine θ

SecθCotθ = Cosθ / SineθCosθ

2. SecθTanθCscθ

Recall

Sec θ = 1/Cos θ

Tan θ = Sine θ / Cos θ

Csc θ = 1/Sine θ

Thus,

SecθTanθCscθ =

1/Cosθ × Sineθ/Cosθ × 1/Sineθ

= Sineθ / Cos²θSineθ

3. Cscθ/Secθ

Recall

Csc θ = 1/Sine θ

Sec θ = 1/Cos θ

Thus,

Cscθ/Secθ = 1/Sine θ ÷ 1/Cos θ

= 1/Sine θ × Cos θ

= Cosθ / Sineθ

4. Cosθ / Secθ

Recall

Sec θ = 1/Cos θ

Cosθ / Secθ = Cosθ ÷ 1/Cosθ

= Cosθ × Cosθ

= Cos²θ

Recall

Cos²θ + Sin²θ = 1

Cos²θ = 1 – Sin²θ

But

1 = Cos²θ + Sin²θ

Thus,

Cos²θ = Cos²θ + Sin²θ – Sin²θ

Therefore,

Cosθ / Secθ = Cos²θ + Sin²θ – Sin²θ