Important Trigonometric Identities
sin2 θ + cos2 θ = 1	1 + tan2θ = sec2θ
cosec2 θ = 1 + cot2 θ	sin 2θ = 2 sinθ cosθ
sin (A+B) = sin A cos B + cos A sin B	cos 2θ = 1 – 2sin2 θ
sin (A-B) = sin A cos B – cos A sin B	tan 2θ = (2tanθ)/(1 – tan2θ)
cos (A+B) = cos A cos B – sin A sin B	​​sin3θ = 3sinθ − 4sin3θ
cos (A-B) = cos A cos B + sin A sin B	cos3θ = 4cos3θ − 3cosθ
tan (A+B) = (tan A + tan B)/(1 – tan A tan B)	tan3θ = (3tanθ − tan3θ​ )/1 – 3tan2θ
tan (A-B) = (tan A – tan B)/(1 + tan A tan B)	sinA + sinB = 2 sin (A+B)/2 cos(A-B)/2
sinA cosB = [sin(A+B) + sin(A−B)​]/2	cosA + cosB = 2 cos(A+B)/2 cos(A-B)/2
cosA cosB = [cos(A+B) + cos(A−B)​]/2	sinA – sinB = 2 cos (A+B)/2 sin(A-B)/2
sinA sinB= [cos(A−B) – cos(A+B)]/2	cosA – cosB = -2 sin(A+B)/2 sin(A-B)/2

Trigonometric Identities Practice Problems

Problem 1: Find the value of [Tex]\frac{\sin x}{1 + \cos x} + \frac{\cos x}{1 + \sin x}[/Tex].

Solution:

To simplify this expression, we can find a common denominator:

[Tex]\frac{\sin x(1+\sin x)+ \cos x(1+\cos x)}{(1+\cos x)(1+\sin x)}[/Tex]

Expanding the numerator, we get

sin x + sin² x + cos x + cos² x

As we know sin² x + cos² x = 1, hence the above equation becomes:

sin x + cos x + 1

Hence the value of given expression is: [Tex]\frac{sin x + cos x + 1} {(1 + cos x)(1+sin x)}[/Tex]

Problem 2: Prove that sin (45° – a) cos (45° – b) + cos (45° – a) sin (45° – b)  = cos (a + b).

Solution:

Let us solve the LHS of the given equation: 

By using formula: sin (A + B) = sin A cos B + cos A sin B we get

sin(45° – a) cos (45° – b) + cos (45° – a) sin (45° – b) = sin [(45°– a) + (45° – b)]

= sin [90° – (a + b)]

As sin (90° – θ) = cos θ, hence

sin [90° – (a + b)] = cos (a + b)

= R. H. S  

∴ LHS = RHS [Hence Proved]

Problem 3: Show that (tan² θ + tan⁴ θ) = (sec⁴ θ – sec² θ)

Solution:

Let us take the RHS of the given equation:

We have sec⁴θ – sec²θ

Take sec²θ common

sec²θ(sec²θ – 1)

We know, sec²θ = 1 + tan²θ, Hence the above equation become:

(1 + tan²θ) (1 + tan²θ – 1)

⇒ (1 + tan²θ) tan²θ

⇒ (tan²θ + tan⁴θ) = LHS      

∴ LHS = RHS [Hence Proved]

Problem 4: Find the value of sin(π/4 – π/6).

Solution:

Given, sin (π/4 – π/6)

By using formula: sin (A – B) = sin A cos B – cos A sin B, we get

sin (π/4 – π/6) = sin π/4 cos π/6 – cos π/4 sin π/6   

Since, cos π/4 = sin π/4 = 1/√2, cos π/6 = √3/2, and sin π/6 = 1/2

Putting these values above we get,

sin (π/4 – π/6) = (1/√2) (√3/2) – (1/√2)(1/2)

= (√3 – 1)/2√2

Hence, sin (π/4 – π/6) = (√3 – 1)/2√2

Problem 5: Solve (1 + tan²θ) cos²θ  

Solution:

Given, (1 + tan²θ)cos²θ

Since we know 1 + tan²θ = sec²θ Hence the above equation becomes:

sec²θ . cos²θ

⇒ (1/cos²θ) . cos²θ = 1

Hence (1 + tan²θ)cos²θ = 1

Practice Problems on Trigonometric Identities

Below are some practice problems on trigonometric identities:

P1. Simplify the expression [Tex]\frac{sin^2}{1-cosx} + \frac{cos^2}{1+sinx}[/Tex].

P2. Prove the identity [Tex]\frac1{sinx \cdot cosx} = \frac1{sinx} + \frac 1{cosx}[/Tex]

P3. Prove the identity [Tex]\frac {\tan x} {1-\cot x} + \frac {\cot x} {1-\tan x}[/Tex]

P4. Simplify the expression [Tex]\frac {\sin^2 x}{\cos x} +\frac {\cos^2 x}{\sin x} [/Tex]

P5. Prove the identity sinx tanx + cosx cotx = 2.

P6. Simplify the expression [Tex]\frac 1{\cos x} \cdot \frac1{1+ \sin x} +\frac 1{\sin x} \cdot \frac1{1+ \cos x} [/Tex]

P7. Evaluate: [Tex]\frac {1 + \tan x}{1 – \tan x} = \frac {1 + \sin x}{1 – \sin x}[/Tex]

P8. Prove the identity sin² x + cos² x = 1

P9. Prove the identity [Tex]\frac{\sin x} {1-\cos x} + \frac{\cos x} {1-\sin x} = \frac 2 {\sin x+ \cos x} [/Tex]

P10. Simplify the expression [Tex]\sin x + \tan x \cdot \cos x + \cot x \cdot \cos x[/Tex]

Read More,

• Trigonometry Table
• Sum and Difference Formulas and Identities in Trigonometry
• Standard Identities Practice Questions
• Trigonometry Questions with Solutions

Frequently Asked Questions

What is Trigonometric Identity?

Trigonometric identities are equations in trigonometry that are always true for all valid values of the angles involved.

Why are trigonometric identities important?

Trigonometric identities are fundamental in various branches of mathematics and sciences, including calculus, physics, engineering, and more. They allow us to simplify complex expressions, solve equations involving trigonometric functions, and understand relationships between angles and sides in geometric contexts like triangles.

What are the fundamental formula of trigonometry?

The three fundamental formulas of trigonometry are: sin²A + cos²A = 1. 1 + tan²A = sec²A. 1 + cot²A = cosec²A

What are some applications of trigonometric identities?

Trigonometric identities have real-world applications:

• Physics: They are used in analyzing periodic motion (e.g., waves, oscillations), projectile motion, and resolving forces into components.
• Engineering: They play a role in various engineering fields like electrical circuits (AC current), structural analysis (forces in beams), and signal processing.
• Computer Graphics: They are essential for 3D graphics, where calculations involving angles and rotations are crucial for rendering objects and animations.