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**>> 5.2. Sum and Difference Formulas**** **

5.1. Verifying Trigonometric Identities

One thing that makes trigonometry so useful is that there are numerous trigonometric identities. In this section, we state and derive a few of the most basic ones and verify them for various values of the arguments.

We have already seen several trigonometric identities in the previous chapter. These were mainly identities which show how various trigonometric functions transform under translations and reflections, such as sin(π/2 - θ) = cos θ and tan(-θ) = -tan θ. Here we derive some other quite useful identities of a different form.

One of the most basic trigonometric identites is the following:

**(5.1.1)**sin^{2}θ + cos^{2}θ = 1

(The notation sin^{2}θ is shorthand for (sin^{ }θ)^{2}, etc.) It is easy to prove this equation. Consider Figure 4.2.1. We see that the point P = (x = cos θ, y = sin θ) lies on the unit circle, so its distance to the origin is 1. But we know from the distance formula (1.3.1) that this distance is also equal to d(O,P) = √x^{2} + y^{2} = √cos^{2}θ + sin^{2}θ = 1, whence the result follows.

Two similar trigonometric identities are easily derived from (5.1.1), namely the following:

**(5.1.2a)**tan^{2}θ + 1 = sec^{2}θ**(5.1.2a)**cot^{2}θ + 1 = csc^{2}θ

Equation (5.1.2a) followes immediately from (5.1.1) by dividing both sides by cos^{2}θ. Similarly, (5.1.2b) follows from (5.1.1) by dividing both sides by sin^{2}θ.

**Example 1:** Verify (5.1.1) for θ = π/6.

**Solution: **We must check that sin^{2}(π/6) + cos^{2}(π/6) = 1. The left side is (√3^{}/2)^{2} + (1/2)^{2} = 3/4 + 1/4 = 1, verifying the identity.

**Example 2:** Verify (5.1.2a) for θ = π/4.

**Solution: **We must check that tan^{2}(π/4) + 1 = sec^{2}(π/4). The left side is 1 + 1 = 2 and the right side is (√2)^{2} = 2, verifying the identity.

**Example 3:** Verify (5.1.2b) for θ = π/3.

**Solution: **We must check that cot^{2}(π/3) + 1 = csc ^{2}(π/3). The left side is (√3^{}/3)^{2} + 1 = 1/3 + 1 = 4/3 and the right side is (2√3/3)^{2} = 12/9 = 4/3, verifying the identity.