Question

Cos 30 degree Cos 60 degree minus sin 30 degree sin 60 degree is equal to cos 90 degree

Answer 1

To Prove

→ cos30° cos60° - sin30° sin60° = cos90°

Proof

What we have?

• cos30° = sin60° = √3/2
• cos60° = sin30° = 1/2
• cos90° = 0

Given

→ R.H.S → cos90° = 0

L.H.S→ cos30° cos60° - sin30° sin60°

By substituting values we get,

→ √3/2 × 1/2 - 1/2 × √3/2

→ √3/4 - √3/4 = 0 = R.H.S

Hence Proved

Answer 2

To prove:

cos30cos60 - sin30sin60=cos90

Let us first find the individual values of the given trigonometric functions

$\sf{cos90 = 0} \\ \sf{sin30 = cos60 = \frac{1}{ \sqrt{2} } } \\ \sf{cos30 = sin60 = \frac{ \sqrt{3} }{2} }$

Now,

sin30sin60 - cos30cos60

$\sf{ = \frac{1}{ \sqrt{2} } \times \frac{ \sqrt{3} }{2} - \frac{1}{ \sqrt{2} } \times \frac{ \sqrt{3} }{2} } \\ \\ = \: \sf{ \frac{ \sqrt{3} }{2 \sqrt{2} } - \frac{ \sqrt{3} }{2 \sqrt{2} }} \\ \\ = \sf{0} \\ \\ \sf{ = cos90}$

Hence,proved