Question

Without using trigonometric tables, prove that:

sec²50° – cot²40° = 1

Answer 1

Given: sec²50° – cot²40°

To Prove: sec²50° – cot²40° = 1

Proof [Step-wise]:

1. Use the Pythagorean identity: sec²θ = 1 + tan²θ. 

Therefore sec²50° = 1 + tan²50°.

2. Use the complementary-angle relation: cot α = tan(90° – α).

Hence cot 40° = tan(90° – 40°) = tan 50°, so cot²40° = tan²50°.

3. Substitute into the expression:

sec²50° – cot²40° = (1 + tan²50°) – tan²50°.

4. Simplify: (1 + tan²50°) – tan²50° = 1.