Trigonometric Identities Practice Problems

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 (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 \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