6.4 Basic Trigonometric Identities

6.4 Basic Trigonometric Identities

Three basic trigonometric identities are:

 sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x cot2 x + 1 = cosec2 x

Example 1 (To Prove Trigonometric Identities which involve the Three Basic Identities)
Prove each of the following trigonometric identities.
(a) sin2 x – cos2 x = 1 – 2 cos2 x
(b) (1 – cosec2 x) (1– sec2 x) = 1

Solution:
(a)
sin2 x– cos2 x = 1 – 2 cos2x
LHS: sin2 x – cos2 x
= 1 – cos2 x – cos2 x
= 1 – 2 cos2 x (RHS)

(b)
$\begin{array}{l}\left(1-{\mathrm{cosec}}^{2}x\right)\left(1-{\mathrm{sec}}^{2}x\right)=1\\ \text{LHS:}\left(1-{\mathrm{cosec}}^{2}x\right)\left(1-{\mathrm{sec}}^{2}x\right)\\ =\left(-{\mathrm{cot}}^{2}x\right)\left(-{\mathrm{tan}}^{2}x\right)\\ =\left({\mathrm{cot}}^{2}x\right)\left({\mathrm{tan}}^{2}x\right)\\ =\left(\frac{1}{{\mathrm{tan}}^{2}x}\right){\mathrm{tan}}^{2}x\\ =1\text{(RHS)}\end{array}$

Example 2 (To Solve Trigonometric Equations which involve the Three Basic Identities)
Solve the following trigonometric equations for 0ox ≤ 360o.
(a) sin2 x cos x + 1 = cos x
(b) 2 cosec2 x – 5 cot x = 0

Solution:
(a)
sin2 cos x + 1 = cos x
(1 – cos2 x) cos x + 1 = cos x
cos x – cos3 x + 1 = cos x
cos3 x = 1
cos x = 1
x = 0o, 360o

(b)
2 cosec2 x – 5 cot x = 0
2 (1 + cot2 x) – 5 cot x = 0
2 + 2 cot2 x – 5 cot x = 0
2 cot2 x – 5 cot x + 2 = 0
(2 cot x – 1) (cot x – 2) = 0
cot x= ½ or cotx = 2
cot x= ½ or cot x = 2
tan x = 2 tan x = ½
x =63.43o, 243.43o   x = 26.57o, 206.57o

(Note: tangent is positive in the first and third quadrants)

Thus, x = 26.57o, 63.43o, 206.57o, 243.43o

4 thoughts on “6.4 Basic Trigonometric Identities”

1. This is really useful, thanks.

2. what is LHS and RHS means>

• LHS = Left Hand Side
RHS = Right Hand Side

3. This is very useful ???