Question

simplify this 8 question ​

Answer 1

Answer:

hello ansh

(Sin P/2 + Cos P/2)2 = Sin2 P/2 + Cos2 P/2 +2Sin P/2Cos P/2

= 1 + sinP

Sin P/2 + Cos P/2 = ± √ (1 + sin P)

If P = 30° so P/2 = 30/2 =15°

Putting this value in the above equation:

Sin 15° + Cos 15° = ±√ (1 + sin 30) …(1)

Also, (Sin P/2 – Cos P/2)2 = Sin2 P/2 + Cos2 P/2 – 2Sin P/2Cos P/2

= 1 – sinP

Sin P/2 – Cos P/2 = ± √(1 – sin P)

Putting this value in the above equation:

Sin 15° – Cos 15° = ±√(1 – sin 30°) …(2)

As seen, sin 15° > 0 and cos 15° > 0

hence, sin 15° + cos 15° > 0

From (1) we will get,

sin 15° + cos 15° = √(1 + sin 30°) …(3)

Also, sin 15° – cos 15° = √2 (1/√2 sin 15˚ – 1/√2 cos 15˚)

or, sin 15° – cos 15° = √2 (cos 45° sin 15˚ – sin 45° cos 15°)

or, sin 15° – cos 15° = √2 sin (15˚ – 45˚)

or, sin 15° – cos 15° = √2 sin (- 30˚)

or, sin 15° – cos 15° = -√2 sin 30°

or, sin 15° – cos 15° = -√2 x 1/2

or, sin 15° – cos 15° = – √2/2

So, sin 15° – cos 15° < 0

Now we got, from (2) sin 15° – cos 15°= -√(1 – sin 30°) … (4)

Adding eq. (3) and (4) we get,

2 sin 15° = √(1 + ½) – √(1 – ½)

2 sin 15° = (√3−1)/√2

∴ sin 15° = (√3−1)/2√2