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)
( 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 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


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