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State true or false.

$sin 30 ° ⋅ cos 45 ° + cos 30 ° ⋅ sin 45 ° = 3 (1 − 2) 4$ .

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True

False

answer is B.

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Detailed Solution

Given trigonometric expression is

s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 °

.
We know that,

sin 30 ° = 12

cos 30 ° = 32

cos 45 ° = s i n 45 ° = 12

Putting the values in the expression, we get,

s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 12 × 12 + 32 × 12

.
Taking common, we get,

⇒ s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 12 12 + 32

⇒ s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 12 1 + 3 2

⇒ s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 1 + 3 22

Rationalizing the denominator, we get,

⇒ s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 1 + 3 22 × 22

⇒ s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 2 1 + 3 4

Thus, the correct statement is that,

s i n 30 ° . c o s 45 ° + c o s 30 ° . s i n 45 ° = 2 1 + 3 4 .

Therefore, the statement is false.
Hence, option 2 is correct.

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