Online Trigonometry Tutoring: Trigonometric Identities

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Identity is an equation which is always true for all defined values

Tangent and cotangent functions can be expressed in terms of the sine and cosine functions

tan x = $\frac{sin x}{cos x}$

cot x = $\frac{cos x}{sin x}$

These reciprocals are used to find the value of trig functions:

sin x = $\frac{1}{cosec x}$ ; cosec x = $\frac{1}{sin x}$

cos x = $\frac{1}{sec x}$ ; sec x = $\frac{1}{cos x}$

tan x = $\frac{1}{cot x}$ ; cot x = $\frac{1}{tan x}$

Pythagorean Identities is to help simplify trigonometric expressions.
Simplifying trigonometric expressions to create a factorable expression used to find missing trigonometric values.
Used to find missing trigonometric values

$sin^2\theta$ + $cos^2\theta$ = 1

$tan^2\theta$ + 1 = $sec^2\theta$

1 + $cot^2\theta$ = $cosec^2\theta$

Example 1

Prove that tan y/sin y = sec y

Expressing it in terms of sin y or cos y, we get

LHS = tan y × 1/sin y

= (sin y/cos y) × 1/sin y (tan A = sin A/cos A)

= 1/cos y

= sec y (since 1/cos A = sec A)

= RHS

Example 2

Verify the identities

1/sec²a = sin²a . cos²a + cos⁴a

RHS = cos²a (sin²a + cos²a)

= cos²a (1) {we know, sin²a + cos²a =1}

= cos²a

LHS = 1/sec²a = 1/(1/ cos²a) = cos²a

Check Point

Problem 1. If sinθ = 2/3, find cosθ.

Problem 2. If cos θ = 5/11 and sin θ = 9/11, what is the value of tan θ?

Problem 3. Express sin 67° + cos 65° in terms of trigonometric ratios of angle between 0 and 45 degrees.

Problem 4. If cos θ = .7, find the value of sec θ.

Answer Key

1. cos θ = $\sqrt{5}/3$
2. tan θ = 9/5
3. cos 23° + sin 25°
4. sec θ = 10/7

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