# 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

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

**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**

**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 Ã— 1/sin y

= (sin y/cos y) Ã— 1/sin y (tan A = sin A/cos A)

= 1/cos y

= sec yÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (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}aÂ = 1/(1/ cos^{2}a) = cos^{2}a

**Check Point**

Problem 1. If sinÎ¸ = 2/3, find cosÎ¸.

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

Problem 3. ExpressÂ sin 67^{0} + cos 65^{0}Â in terms of trigonometric ratios of angleÂ between 0 and 45 degrees.

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

Answers:

1. cos Î¸ = ïƒ–5/3

2. tan Î¸ = 9/5

3. cos 23^{0} + sin 25^{0}

4. sec Î¸ = 10/7

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