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The product of sin 30 and sin 60 is same as the product of cos 60 degrees tan 30 degrees and cos 30 degrees tan 60 degrees.

Solution:

We have to prove that the product of sin 30 and sin 60 is the same as the product of cos 60 degrees tan 30 degrees and cos 30 degrees tan 60 degrees.

LHS:

sin 30° = 1/2

sin 60° = √3/2

Product of sin 30° and sin 60° = (1/2)(√3/2)

= √3/4

RHS:

cos 60° = 1/2

tan 30° = 1/√3

Product of cos 60° and tan 30° = (1/2)(1/√3)

= 1/2√3

cos 30° = √3/2

tan 60° = √3

Product of cos 30° and tan 60° = (√3/2)(√3)

= 3/2

Now, product of cos 60° and tan 30° and cos 30° and tan 60° = (1/2√3)(3/2)

= (√3 × √3)/(2√3 × 2)

= √3/4

LHS = RHS

Therefore, the product of sin 30 and sin 60 is the same as the product of cos 60 degrees tan 30 degrees and cos 30 degrees tan 60 degrees.

Summary:

The product of sin 30 and sin 60 is the same as the product of cos 60 degrees tan 30 degrees and cos 30 degrees tan 60 degrees.

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