Question

Without using tables verify that cos 60° = cos² 30° – sin² 30°.

Answer 1

Given: cos 60° = cos² 30° – sin² 30°.

Step-wise calculation:

1. Use the double-angle identity cos (2θ) = cos²θ – sin²θ.

2. Put θ = 30°:

cos 60° = cos² 30° – sin² 30°   ...(This is the identity specialized to 30°)

3. Evaluate the basic values:

`cos 30^circ = sqrt(3)/2` and `sin 30^circ = 1/2`

4. Compute:

cos² 30° – sin² 30°

= `(sqrt(3)/2)^2 - (1/2)^2`

= `3/4 - 1/4`

= `1/2`

5. Note `cos 60^circ = 1/2`, matching the result.

`cos 60^circ = 1/2 = cos^2 30^circ - sin^2 30^circ`, so the identity is verified.