Proving Trigonometric Ratios

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

1. If sin^4x + sin² x = 1 then proof that, cot^4x + cot² x = 1.

Solution:

Given, sin^4x + sin² x = 1

⇒ sin^4x + sin² x - sin² x = 1 - sin² x, [subtract sin² x from both the sides]

⇒ sin^4x = 1 - sin² x

⇒ sin^4x = cos² x.



Now L.H.S. = cot^4x + cot² x

= cos^4x/sin^4x + cos² x/sin² x

= cos^4x/cos² x + sin^4x/sin² x, [since, sin^4x = cos² x and cos² x = sin^4x]

= 1 = R.H.S. [since we know, sin² x + cos² x = 1]

                                                                              (Proved)

 

2. If sin θ - cos θ = √2 cos θ then proof that sin θ + cos θ = √2 sin θ, where 0 < θ < π/2

Solution:

Given, sin θ - cos θ = √2 cos θ

⇒ (sin θ - cos θ)² = (√2 cos θ)², [squaring both the sides]

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

⇒ sin² θ + cos² θ - 2 sin θ cos θ - cos² θ = 2 cos² θ - cos² θ, [subtract cos² θ from both the sides]

⇒ sin² θ - 2 sin θ cos θ = cos² θ

⇒ sin² θ - 2 sin θ cos θ + 2 sin θ cos θ = cos² θ + 2 sin θ cos θ, [adding 2 sin θ cos θ on both the sides]

⇒ sin² θ = cos² θ + 2 sin θ cos θ

⇒ sin² θ + sin² ϴ = sin² θ + cos² θ + 2 sin θ cos θ, [adding sin² θ on both the sides]

⇒ 2 sin² θ = (sin θ + cos θ)²

⇒ (sin θ + cos θ)² = 2 sin² θ

Now taking square root on both the sides we get,

⇒ sin θ + cos θ = ± √2 sin θ

According to the question, 0 < θ < π/2, hence we neglect the negative vaue.

Therefore, sin θ + cos θ = √2 sin θ

                                                           (Proved)

The above explanation on proving trigonometric ratios will help us to solve different types of trigonometric problems.

● Trigonometric Functions

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit 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
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

10th Grade Math

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