Question

Prove that sin 15° is equal to√6−√2/4​

Answer 1

Step-by-step explanation:

We will use the identity sin(x−y)=sinxcosy−sinycosx. We have that

sin15∘=sin(45−30)∘=sin45∘cos30∘−cos45∘sin30∘=12⎯⎯√3⎯⎯√2−12⎯⎯√12=2⎯⎯√3⎯⎯√2×2−2⎯⎯√2×2=6⎯⎯√−2⎯⎯√4.

We now have

cos215∘=1−sin215∘=1−6+2−26⎯⎯√2⎯⎯√16=8+212⎯⎯⎯⎯√16=6+2+22⎯⎯√6⎯⎯√16=(6⎯⎯√+2⎯⎯√)242,

and so, since cosθ is positive between 0∘ and 90∘,

cos15∘=6⎯⎯√+2⎯⎯√4.

Finally, we have

tan15∘=sin15∘cos15∘=6⎯⎯√−2⎯⎯√6⎯⎯√+2⎯⎯√=(6⎯⎯√−2⎯⎯√)2(6⎯⎯√+2⎯⎯√)(6⎯⎯√−2⎯⎯√)=6+2−22⎯⎯√6⎯⎯√6−2=8−212⎯⎯⎯⎯√4=2−3⎯⎯√.

Answer 2

Answer:

√6 - √2

4

Step-by-step explanation:

sin 15 = sin(45-30)

$sin(a - b) = sina \: cosb - cosa \: sinb$

= sin45 cos30 - cos45 sin30

= (1/√2)(√3/2) - (1/√2)(1/2)

= 1/√2[(√3/2)-(1/2)]

= √3-1/ 2√2

= [(√3-1)/2√2](√2/√2)

= √6-√2/4