Question

Simply (165x165) +2 x165x35+35x35 using identites

Answer 1

${165}^{2} + 2 \times 165 \times 35 + {35}^{2}$
So,
Using
${(x + y)}^{2}$
It will be
${(165 + 35)}^{2}$
${200}^{2}$
Which is = 40000

Answer 2

Call tan 165 = tan t --> tan 2t = tan 330
Trig table and unit circle give:.
tan
330
=
tan
(
−
30
+
360
)
=
tan
(
−
30
)
=
−
tan
30
=
−
1
√
3

Use trig identity:
tan
2
t
=
2
tan
t
1
−
tan
2
t

In this case :
−
1
√
3
=
2
tan
t
1
−
tan
2
t

tan
2
t
−
1
=
2
√
3
tan
t

tan
2
t
−
2
√
3
tan
t
−
1
=
0

Solve this quadratic equation for tan t using improved quadratic formula (Socratic Search):
D
=
d
2
=
b
2
−
4
a
c
=
12
+
4
=
16
-->
d
=
±
4
.
There are 2 real roots:
#tan t = - b/(2a) +- d/(2a) = sqrt3 +- 2
tan t = sqrt3 + 2, and tan t = sqrt3 - 2
Since (165) is in Quadrant II, tan 165 is negative, then take the negative value.
tan (165) = tan t = sqrt3 - 2