Trigonometric Identities Practice Problems (original) (raw)

Last Updated : 23 Jul, 2025

Trigonometric identities are a set of formulas that can be used to reduce a variety of complex equations that contain trigonometric functions. These identities connect the various trigonometric functions – sine (sin), cosine (cos), tangent (tan), and their reciprocals (cotangent, secant, cosecant).

In this article, we will list some of the basic trigonometric identities and solve a few questions based on them. This article will also provide a few unsolved questions to practice.

List of Basic Trigonometric Identities

Below is a list of a few important trigonometric identities:

Trigonometric Identities Practice Problems

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

**Solution:

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

\frac{\sin x(1+\sin x)+ \cos x(1+\cos x)}{(1+\cos x)(1+\sin x)}

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: \frac{sin x + cos x + 1} {(1 + cos x)(1+sin x)}

**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 (tan 2 **θ + tan 4 **θ) = (sec 4 **θ – sec 2 **θ)

**Solution:

Let us take the RHS of the given equation:

We have _sec_4__θ – sec_2__θ

Take sec2θ common

sec2θ(sec2θ – 1)

We know, _sec_2__θ = _1 + tan_2__θ, Hence the above equation become:

_(1 + tan_2__θ) (1 + tan_2__θ – 1)

_⇒ (1 + tan_2__θ) tan_2__θ

⇒ (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 + tan 2 θ) cos 2 θ

**Solution:

Given, (1 + tan2θ)cos2θ

Since we know _1 + tan_2__θ = sec2θ Hence the above equation becomes:

_sec_2__θ . cos_2__θ

⇒ (1/cos2θ) . cos2θ = 1

Hence _(1 + tan_2__θ)cos_2__θ = 1

Practice Problems on Trigonometric Identities

Below are some practice problems on trigonometric identities:

**P1. Simplify the expression \frac{sin^2}{1-cosx} + \frac{cos^2}{1+sinx}.

**P2. Prove the identity \frac1{sinx \cdot cosx} = \frac1{sinx} + \frac 1{cosx}

**P3. Prove the identity \frac {\tan x} {1-\cot x} + \frac {\cot x} {1-\tan x}

**P4. Simplify the expression \frac {\sin^2 x}{\cos x} +\frac {\cos^2 x}{\sin x}

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

**P6. Simplify the expression \frac 1{\cos x} \cdot \frac1{1+ \sin x} +\frac 1{\sin x} \cdot \frac1{1+ \cos x}

**P7. Evaluate: \frac {1 + \tan x}{1 - \tan x} = \frac {1 + \sin x}{1 - \sin x}

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

**P9. Prove the identity \frac{\sin x} {1-\cos x} + \frac{\cos x} {1-\sin x} = \frac 2 {\sin x+ \cos x}

**P10. Simplify the expression \sin x + \tan x \cdot \cos x + \cot x \cdot \cos x

**Read More,

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