Question

Without using trigonometric tables, prove that:

cos²72° + cos²18° = 1

Answer 1

Given: cos²72° + cos²18°

To Prove: cos²72° + cos²18° = 1

Proof [Step-wise]:

1. Note that 72° = 90° – 18°

So, cos 72° = cos (90° – 18°) = sin 18°   ...(Complementary-angle identity)

2. Substitute into the expression:

cos²72° + cos²18° = sin²18° + cos²18°

3. Use the Pythagorean identity sin²θ + cos²θ = 1 to get sin²18° + cos²18° = 1.

4. Therefore cos²72° + cos²18° = 1.