Question

Cos 15 - sin 15 = 1/2 prove that :​

Answer 1

Answer:

Cos(15) = Cos(45-30)

Sin(15) = Sin(45-30)

Cos(45-30) - Sin(45-30)

Simplifying both brackets

Recall

Cos(A+B) = CosACosB - SinASinB (For cosine... The sign is always the opposite during expansion. If you look... Cos(A+B) became negative during its expansion;So if It was Cos(A–B)... Its expansion is CosACosB + SinASinB)]

Sin(A+B) = SinACosB + CosASinB

Now let's go!

Cos(45-30) = Cos45Cos30 + Sin45Sin30

From Trig

Cos45 = 1/√2

Cos30 = √3/2

Sin45 = 1/√2

Sin30 = 1/2

Substituting

Cos(45-30) = (1/√2).(√3/2) + (1/√2).(1/2)

= √3/(2√2) + 1/(2√2)

= 1 + √3/(2√2).

For

Sin(45 - 30)

= Sin45cos30 – Cos45Sin30

= (1/√2).(√3/2) – (1/√2).(1/2)

= √3/(2√2) – 1/2√2

= √3 – 1/(2√2)

So the question was

Cos15 - Sin15

Substituting...

1 + √3 / 2√2 – (√3 - 1)/ 2√2

When the Minus interacts with the parenthesis

We have

1 + √3 - √3 + 1 / (2√2)

= 2/2√2

= 1/√2.

THE ANSWER IS 1/√2 AND NOT 1/2.

YOU CAN VERIFY THIS WITH YOUR CALCULATOR ALSO.

YOU'LL HAVE 0.7071 WHICH IS SAME AS 1/√2.

HAVE A GREAT DAY!