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) sin²x – cos² x = 1 – 2 cos² x

(b) (1 – cosec²x) (1– sec² x) = 1

Solution:

(a)

sin²x– cos² x = 1 – 2 cos²x

LHS: sin²x – cos² x

= 1 – cos² x – cos² x

= 1 – 2 cos² x (RHS)

(b)

\begin{array}{l} (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) \\ = (\frac{1}{\tan^{2} x}) \tan^{2} x \\ = 1 (RHS) \end{array}

Example 2 (To Solve Trigonometric Equations which involve the Three Basic Identities)

Solve the following trigonometric equations for 0^o≤ x ≤ 360^o.

(a) sin²x cos x + 1 = cos x

(b) 2 cosec²x – 5 cot x = 0

Solution:

(a)

sin²x cos x + 1 = cos x

(1 – cos²x) cos x + 1 = cos x

cos x – cos³x + 1 = cos x

cos³x = 1

cos x = 1

x = 0^o, 360^o

(b)

2 cosec²x – 5 cot x = 0

2 (1 + cot²x) – 5 cot x = 0

2 + 2 cot²x – 5 cot x = 0

2 cot²x – 5 cot x + 2 = 0

(2 cotx – 1) (cot x – 2) = 0

cotx= ½ or cotx = 2

tan x = 2 tan x = ½

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

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

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