Maths

190 MATHEMATICS
6. If A, B and C are interior angles of a triangle ABC, then show that
= ✁
✂ ✄ ✝
☎ ✆
sin
B+C
2
A
cos 2
7. Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.
8.5 Trigonometric Identities
You may recall that an equation is called an identity
when it is true for all values of the variables involved.
Similarly, an equation involving trigonometric ratios
of an angle is called a trigonometric identity, if it is
true for all values of the angle(s) involved.
In this section, we will prove one trigonometric
identity, and use it further to prove other useful
trigonometric identities.
In ✞ ABC, right-angled at B (see Fig. 8.22), we have:
Dividing each term of (1) by AC 2 , we get
Fig. 8.22
AB 2 + BC 2 =AC 2 (1)
AB
AC
2 2
BC
✟ =
AC
2 2
AC
AC
2
2
i.e.,
2 2
AB ✠ BC ✡ ✠ ✡
☛ ☞ ✌ ☞ ✌
AC AC
✍ ✎ ✍ ✎
=
AC ✠ ✡
☞ ✌
AC
✍
✎
2
i.e., (cos A) 2 + (sin A) 2 =1
i.e., cos 2 A + sin 2 A = 1 (2)
This is true for all A such that 0° ✏ A ✏ 90°. So, this is a trigonometric identity.
Let us now divide (1) by AB 2 . We get
AB
AB
2 2
BC
✟ =
AB
2 2
AC
AB
2
2
or,
2 2
AB ✠ BC ✡ ✠ ✡
☛ ☞ ✌ ☞ ✌
✍ ✎ ✍ ✎
AB AB
=
AC ✠ ✡
☞ ✌
✍ AB
✎
i.e., 1 + tan 2 A = sec 2 A (3)
2