Verify Trigonometric Identities

How to verify Trigonometric Identities?

To proof and verify the identities we will make use of the basic trigonometric identities to make sure that both the sides of the equation is equal to each other.

1. If tan A = (sin θ - cos θ)/(sin θ + cos θ) then prove that, 
                         sin θ + cos θ  = ± √2 cos A

Solution:

We know that, sec² A = 1 + tan² A

⇒ sec² A = 1 + (sin θ - cos θ)²/(sin θ + cos θ) ²

⇒ sec² A = [(sin θ + cos θ) ² + (sin θ - cos θ) ²]/(sin θ + cos θ) ²

⇒ sec² A = 2(sin² θ + cos² θ)/ (sin θ + cos θ) ²



⇒ 1/cos² A = 2/(sin θ + cos θ) ²

⇒ (sin θ + cos θ) ² = 2 cos²

Now taking square root on both the sides we get,

sin θ + cos θ =  ± √2 cos A .

                                                  Proved

More examples to get the basic ideas to proof and verify Trigonometric Identities.

2. If x sin³ θ + y cos³ θ = sin θ cos θ and x sin θ – y cos θ = 0, then prove that x² + y² = 1, (where, sin θ ≠ 0 and cos θ ≠ 0).

Solution:

x sin θ - y cos θ = 0, (Given)

⇒ x sin θ = y cos θ

⇒ y cos θ = x sin θ

Now dividing both sides by cos θ we get,

y = x ∙ (sin θ/cos θ)

Again, x sin³ θ + y cos³ θ = sin θ cos θ

⇒ x sin³ θ + x ∙ (sin θ /cos θ) ∙ cos³ θ = sin θ cos θ [Since, y = x ∙ (sin θ/cos θ)]

⇒ x sin θ ( sin² θ + cos² θ) = sin θ cos θ, [since, cos θ ≠ 0]

⇒ x sin θ (1) = sin θ cos θ,[since, sin² θ + cos² θ = 0]

⇒ x sin θ = sin θ cos θ

Now dividing both sides by sin θ we get,

⇒ x = cos θ, [since, sin θ ≠ 0]

Therefore, y = x ∙ (sin θ/cos θ)

⇒ y = cos θ ∙ (sin θ/cos θ), [Putting x = cos θ]

⇒ y = sin θ

Now, x² + y²

= cos² θ + sin² θ

= 1.

Therefore, x² + y² = 1.

                                                  Proved

3. If 2y cos α = x sin α and 2x sec α - y csc α = 3, then prove that x² + 4y² = 4

Solution:

2y cos α = x sin α, (Given)

\(\frac{cos  α}{x}  =  \frac{sin  α}{2y} = \frac{\sqrt{cos^{2} α  +  sin^{2} α}}{x^{2}  +  4y^{2}} = \frac{1}{x^{2}  +  4y^{2}}
\)

\(Therefore, cos  θ  =  \frac{x}{x^{2}  +  4y^{2}} and  sin  θ = \frac{2y}{x^{2}  +  4y^{2}}\)

Now, 2x sec α - y csc α = 3

⇒ 2x ∙ \(\frac{1}{cos  α}\) - y ∙ \(\frac{1}{sin  α}\) = 3, [Since, sec α = \(\frac{1}{cos  α}\) and csc α = \(\frac{1}{sin  α}] \)

⇒ 2x ∙ \(\frac{\sqrt{x^{2}  +  4y^{2}}}{x}\) - y ∙ \(\frac{\sqrt{x^{2}  +  4y^{2}}}{2y}\) = 3, [putting the values of sin α and cos α]

⇒ \(\frac{3}{2}\sqrt{x^{2}  +  4y^{2}} = 3\)

⇒ \(\sqrt{x^{2}  +  4y^{2}} = 2\)

Now taking square root on both the sides we get,

⇒ x² + 4y² = 4.

                                                  Proved

Note: Remember there is no set method that can be applied to verify trigonometric identities. However, a few different techniques needed to follow to start verifying from one side, based on the identity which is to be verified.

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● 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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