simplify cos 15°- sin 15° / sin 15° + cos 15°
Answers
Answer:
Step-by-step explanation:
Value of sin 15 degrees
Method 1 ( using sin 30)
(
sin
A
2
+
cos
A
2
)
2
=
sin
2
A
2
+
cos
2
A
2
+
2
sin
A
2
cos
A
2
Now
sin
2
A
2
+
cos
2
A
2
and
sin
A
=
2
sin
A
2
cos
A
2
Therefore
(
sin
A
2
+
cos
A
2
)
2
=
1
+
sin
A
sin
A
2
+
cos
A
2
=
±
√
1
+
sin
A
Similarly
(
sin
A
2
–
cos
A
2
)
2
=
sin
2
A
2
+
cos
2
A
2
–
2
sin
A
2
cos
A
2
or
sin
A
2
–
cos
A
2
=
±
√
1
–
s
i
n
A
Now putting A= 30, we get
sin
15
+
cos
15
=
±
√
1
+
s
i
n
30
and
sin
15
–
cos
15
=
±
√
1
–
s
i
n
30
Now we know that sin 15 > 0 and cos 15 > 0,
Therefore
sin
15
+
cos
15
=
√
1
+
sin
30
=
√
3
√
2
-(1)
But we are not sure about the values of sin 15 – cos 15
Lets see how to determine it
sin
15
–
cos
15
=
√
2
(
1
√
2
s
i
n
15
–
1
√
2
cos
15
)
=
√
2
(
cos
45
sin
15
–
sin
45
cos
15
)
=
√
2
sin
(
15
−
45
)
=
–
√
2
sin
30
So it is a negative
sin
15
–
cos
15
=
–
√
1
–
sin
30
=
–
1
√
2
-(2)
Adding (1) and (2), we get
2
sin
15
=
√
3
√
2
–
1
√
2
sin
15
=
√
3
−
1
2
√
2
Method -2 (using 45 and 60 values)
sin
15
=
sin
(
60
−
45
)
sin
(
A
−
B
)
=
sin
A
cos
B
–
cos
A
sin
B
=
sin
60
cos
45
–
cos
60
sin
45
=
√
3
2
1
√
2
–
1
2
1
√
2
sin
15
=
√
3
−
1
2
√
2
Value of cos 15 degrees
Subtracting (2) from (1), wet get
2
cos
15
=
√
3
√
2
+
1
√
2
cos
15
=
√
3
+
1
2
√
2
Method 2
cos
15
=
cos
(
60
−
45
)
cos
(
A
–
B
)
=
cos
A
cos
B
+
sin
A
sin
B
=
cos
60
cos
45
+
sin
60
sin
45
=
√
3
+
1
2
√
2
Value of tan 15 degrees
Now
tan
15
=
sin
15
cos
15
=
√
3
−
1
√
3
+
1
Value of sin 75 degrees
sin
75
=
sin
(
90
−
15
)
=
cos
15
sin
75
=
√
3
+
1
2
√
2
Value of cos 75 degrees
cos
75
=
cos
(
90
−
15
)
=
sin
15
cos
75
=
√
3
−
1
2
√
2
Value of tan 75 degrees
Now
tan
75
=
sin
75
cos
75
=
√
3
+
1
√
3
−
1