5.4 Basic Trigonometric Identities

5.4 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 cos2 x
LHS: sin2 x – cos2 x
= 1 – cos2 x – cos2 x
= 1 – 2 cos2 x (RHS)

(b)
(1 cosec 2 x)(1 sec 2 x)=1 LHS: (1 cosec 2 x)(1 sec 2 x) =( cot 2 x)( tan 2 x) =( cot 2 x )( tan 2 x ) =( 1 tan 2 x ) tan 2 x =1 (RHS)


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 x 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 cot x = 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