# Online Trigonometry Tutoring: Trigonometric Identities

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**Learn Trigonometric Identities from certified Online Trigonometry tutor:**

**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 =

cot x =

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

sin x = ; cosec x =

cos x = ; sec x =

tan x = ; cot 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**

+ = 1

+ 1 =

1 + =

**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/sec^{2}a = sin^{2}a . cos^{2}a + cos^{4}a

RHS = cos^{2}a (sin^{2}a + cos^{2}a)

= cos^{2}a (1) {we know, sin^{2}a + cos^{2}a =1}

= cos^{2}a

LHS = 1/sec^{2}a = 1/(1/ cos^{2}a) = cos^{2}a

**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.

Answers:

1. cos θ =

2. tan θ = 9/5

3. cos 23° + sin 25°

4. sec θ = 10/7

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