# Elimination of Trigonometric Ratios

Here we will learn about the elimination of trigonometric ratios with the help of different types of problems.

In order to eliminate the T-ratios from the given relations, we make use of the fundamental trigonometrical identities, in the following examples.

Worked-out examples on elimination of trigonometric ratios:

1. If sin θ + sin2 θ = 1, prove that cos2 θ + cos4 θ = 1

Solution:

sin θ + sin2 θ = 1

⇒ sin θ = 1 - sin2 θ, [subtract sin2 θ from both the sides]

⇒ sin θ = cos2 θ, [since, 1 – sin2 θ = cos2 θ]

⇒ sin2 θ = cos4 θ, [squaring both the sides]

⇒ 1 - cos2 θ = cos4 θ, [since sin2 θ = 1 – cos2 θ]

⇒ 1 = cos4 θ + cos2 θ, [adding cos2 θ on both the sides]

⇒ cos4 θ + cos2 θ = 1

Therefore, cos2 θ + cos4 θ = 1

2. If (cos θ + sin θ) = √2 cos θ, shown that (cos θ - sin θ) = √2 sin θ

Solution:

(cos θ + sin θ) = √2 cos θ ………… (A)

⇒ (cos θ + sin θ) 2 = 2 cos2 θ, [squaring both the sides]

⇒ cos2 θ + sin2 θ + 2 sin θ cos θ = 2 cos2 θ

⇒ 2 sin θ cos θ = 2 cos2 θ - cos2 θ - sin2 θ

⇒ 2 sin θ cos θ = cos2 θ - sin2 θ

⇒ cos2 θ - sin2 θ = 2 sin θ cos θ

⇒ (cos θ + sin θ) (cos θ - sin θ) = 2 sin θ cos θ

⇒ (√2 cos θ) (cos θ - sin θ) = 2 sin θ cos θ ………… using (A)

⇒ (cos θ - sin θ) = (2 sin θ cos θ)/(√2 cos θ)

⇒ (cos θ - sin θ) = √2 sin θ

Therefore, (cos θ - sin θ) = √2 sin θ

3. If 3 sin θ + 5 cos θ = 5, prove that (5 sin θ - 3 cos θ) = ± 3.

Solution:

(3 sin θ + 5 cos θ)2 + (5 sin θ - 3 cos θ)2

= (9 sin2 θ + 25 cos2 θ + 30 sin θ cos θ) + (25 sin2 θ                                   + 9 cos2 θ - 30 sin θ cos θ)

= 34 sin2 θ + 34 cos2 θ

= 34 (sin2 θ + cos2 θ)

= 34 (1)

= 34

⇒ (3 sin θ + 5 cos θ)2 + (5 sin θ - 3 cos θ)2 = 34

⇒ (5)2 + (5 sin θ - 3 cos θ)2 = 34, [since, (3 sin θ + 5 cos θ) = 5]

⇒ 25 + (5 sin θ - 3 cos θ)2 = 34

⇒ (5 sin θ - 3 cos θ)2 = 9 [subtract 25 from both the sides]

⇒ (5 sin θ - 3 cos θ) = ± 3

Therefore, (5 sin θ - 3 cos θ) = ± 3.

The above problems on elimination of trigonometric ratios are explained step-by-step so, that students get the clear concept how to make use of the fundamental trigonometrical identities.

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Trigonometric Functions