Trig Ratios Proving Problems

In trig ratios proving problems we will learn how to proof the questions step-by-step using trigonometric identities.

1. If (1 + cos A)( 1 + cos B)( 1 + cos C) = (1 - cos A)( 1 - cos B)( 1 - cos C) then prove that each side = ± sin A sin B sin C.

Solution: Let, (1 + cos A) (1 + cos B) (1 + cos C) = k …. (i)

Therefore, according to the problem,

(1 - cos A) (1 - cos B) (1 - cos C) = k ….. (ii)

Now multiplying both sides of (i) and (ii) we get,

(1 + cos A)(1 + cos B)(1 + cos C)(1 - cos A)(1 - cos B)(1 - cos C) = k²

⇒ k² = (1 - cos² A) (1 - cos² B) (1 - cos² C)

⇒ k² = sin² A sin² B sin² C

k = ± sin A sin B sin C.

Therefore, each side of the given condition

= k = ± sin A sin B sin C
Proved.

More solved examples on trig ratios proving problems.

2. If u_n = cosⁿ θ + sinⁿ θ then prove that, 2u₆ - 3u₄ + 1 = 0.

Solution:

Since, u_n = cosⁿ θ + sinⁿ θ

Therefore, u₆ = cos⁶ θ + sin⁶ θ

⇒ u₆ = (cos² θ)³ + (sin² θ)³

⇒ u₆ = (cos² θ + sin² θ)³ - 3 cos² θ ∙ sin² θ (cos² θ + sin² θ)

⇒ u₆ = 1 - 3cos² θ sin² θ and u₄ = cos⁴ θ + sin⁴ θ

⇒ u₄ = (cos² θ)² + (sin² θ)²

⇒ u₄ = (cos² θ + sin² θ)² - 2 cos² θ sin² θ

⇒ u₄ = 1 - 2 cos² θ sin² θ

Therefore,

2u₆ - 3u₄ + 1

= 2(1 - 3cos² θ sin² θ) - 3(1 - 2 cos² θ sin² θ) + 1

= 2 - 6 cos² θ sin² θ - 3 + 6 cos² θ sin² θ + 1

= 0.

Therefore, 2u₆ - 3u₄ + 1 = 0.

Proved.

3. If a sin θ - b cos θ = c then prove that, a cos θ + b sin θ = ± √(a² + b² - c²).

Solution:

Given: a sin θ - b cos θ = c

⇒ (a sin θ - b cos θ)² = c², [Squaring both sides]

⇒ a² sin² θ + b² cos² θ - 2ab sin θ cos θ = c²

⇒ - a² sin² θ - b² cos² θ + 2ab sin θ cos θ = - c²

⇒ a² - a² sin² θ + b² - b² cos² θ + 2ab sin θ cos θ = a² + b² - c²

⇒ a²(1 - sin² θ) + b²(1 - cos² θ) + 2ab sin θ cos θ = a² + b² - c²

⇒ a² cos² θ + b² sin² θ + 2 ∙ a cos θ ∙ b sin θ = a² + b² - c²

⇒ (a cos θ + b sin θ)² = a² + b² - c²

Now taking square root on both the sides we get,

⇒ a cos θ + b sin θ = ± √(a² + b² - c²).

Proved.

The above three trig ratios proving problems will help us to solve more basic problems on T-ratio.

Basic Trigonometric Ratios

Relations Between the Trigonometric Ratios

Problems on Trigonometric Ratios

Reciprocal Relations of Trigonometric Ratios

Trigonometrical Identity

Problems on Trigonometric Identities

Elimination of Trigonometric Ratios

Eliminate Theta between the equations

Problems on Eliminate Theta

Trig Ratio Problems