Prove that Cos²θ (1+tan²θ )=1

Solution

We can start with the identity:

1 + tan²θ = sec²θ

Dividing both sides by sec²θ, we get:

1/sec²θ + tan²θ/sec²θ = 1

Using the identity sec²θ = 1/cos²θ, we can rewrite this as:

cos²θ/(cos²θ)² + sin²θ/(cos²θ)² = 1

Simplifying the left-hand side using the identity sin²θ + cos²θ = 1, we get:

1/(cos²θ)² = 1

Multiplying both sides by (cos²θ)², we get:

1 = (cos²θ)²

Expanding the left-hand side, we get:

cos²θ * (1 + tan²θ) = 1

Multiplying both sides by cos²θ, we get:

cos⁴θ * (1 + tan²θ) = cos²θ

Dividing both sides by cos⁴θ, we get:

1 + tan²θ = 1/cos²θ

Multiplying both sides by cos²θ, we get:

cos²θ + cos²θ tan²θ = 1

Subtracting cos²θ tan²θ from both sides, we get:

cos²θ (1 + tan²θ) = 1

Therefore, we have shown that Cos²θ (1 + tan²θ) = 1.