Trigonometric Identities Practice Problems and Answers

Why You Need to Practice Trigonometric Identities

Trigonometric identities aren't optional knowledge. They're the foundation for solving calculus problems, physics equations, and engineering calculations. If you can't manipulate identities fluently, you'll hit a wall every time sin²θ + cos²θ appears—which is constantly.

Most students memorize the formulas and still fail. Why? Because memorization without practice is useless. You need to see the patterns, recognize when to apply which identity, and build the muscle memory to transform expressions quickly.

This guide gives you the problems you actually need to work through. No fluff. No explanations of why math matters. Just practice and answers.

The Core Identities You Must Know

Before touching the problems, make sure these are automatic. If you have to think about them, you won't finish timed tests.

Reciprocal Identities

csc θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ

Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Quotient Identities

tan θ = sin θ/cos θ
cot θ = cos θ/sin θ

Co-Function Identities

sin(90° - θ) = cos θ
cos(90° - θ) = sin θ
tan(90° - θ) = cot θ

Double Angle Identities

sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ - sin²θ
cos 2θ = 2cos²θ - 1
cos 2θ = 1 - 2sin²θ

Practice Problems and Solutions

Problem 1: Simplify sin²θ · sec²θ

Solution:

Start by converting sec²θ to its reciprocal form.

sin²θ · sec²θ

= sin²θ · (1/cos²θ)

= sin²θ / cos²θ

= tan²θ

That's it. Recognize when to replace sec with 1/cos.

Problem 2: Simplify (1 - sin²θ) / cos²θ

Solution:

1 - sin²θ is the Pythagorean identity inverted.

(1 - sin²θ) / cos²θ

= cos²θ / cos²θ

= 1

Don't overthink this one. The answer drops out immediately.

Problem 3: Prove that tan θ · cos θ = sin θ

Solution:

Replace tan θ with its quotient identity.

tan θ · cos θ

= (sin θ / cos θ) · cos θ

= sin θ · (cos θ / cos θ)

= sin θ

Left side equals right side. QED.

Problem 4: Simplify sin(90° - θ) + cos(180° - θ)

Solution:

Apply co-function and supplementary angle identities.

sin(90° - θ) = cos θ

cos(180° - θ) = -cos θ

cos θ + (-cos θ) = 0

Problem 5: Find the exact value of cos 75°

Solution:

75° = 45° + 30°. Use the cosine sum formula.

cos(A + B) = cos A cos B - sin A sin sin B

cos 75° = cos 45° cos 30° - sin 45° sin 30°

= (√2/2)(√3/2) - (√2/2)(1/2)

= √6/4 - √2/4

= (√6 - √2)/4

Problem 6: Simplify (sin θ + cos θ)²

Solution:

Expand the binomial, then apply the Pythagorean identity.

(sin θ + cos θ)²

= sin²θ + 2 sin θ cos θ + cos²θ

= (sin²θ + cos²θ) + 2 sin θ cos θ

= 1 + 2 sin θ cos θ

= 1 + sin 2θ

Problem 7: Prove (1 + tan²θ) - sin²θ = sec²θ - sin²θ

Solution:

Both sides simplify to the same expression.

Left side: 1 + tan²θ - sin²θ

= sec²θ - sin²θ

Right side: sec²θ - sin²θ

Both equal sec²θ - sin²θ. The identity holds.

Problem 8: Simplify (sec θ - tan θ)(sec θ + tan θ)

Solution:

This is a difference of squares problem.

(sec θ - tan θ)(sec θ + tan θ)

= sec²θ - tan²θ

Use the Pythagorean identity rearranged: 1 + tan²θ = sec²θ

sec²θ - tan²θ

= (1 + tan²θ) - tan²θ

= 1

Common Mistakes to Avoid

Confusing signs: cos(90° - θ) = sin θ, but sin(90° - θ) = cos θ. The function stays the same.
Forgetting the double angle formula has three forms: cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ. Pick the version that fits your problem.
Multiplying instead of dividing: When you see 1/sin, that's csc. When you see 1/cos, that's sec. Don't flip them.
Skipping steps: Always convert everything to sin and cos first. It makes patterns visible.

How to Approach Any Identity Problem

Step 1: Identify what you're given and what you need

Look at the target expression. Is it all sin and cos? Does it have sec or csc? This tells you which direction to go.

Step 2: Convert everything to sin and cos

Sec, csc, and cot are distractions. Replace them with their reciprocal forms immediately.

Step 3: Expand anything squared

sin²θ becomes (sin θ)². If you see (a + b)², expand it. Algebra is your friend here.

Step 4: Look for Pythagorean patterns

sin²θ + cos²θ, 1 + tan²θ, 1 + cot²θ. These are your escape routes. When you see them, replace with 1 or the equivalent identity.

Step 5: Factor if needed

Sometimes factoring reveals a cancellation. Don't force it, but watch for common factors after expanding.

Quick Reference: When to Use Which Identity

Situation	Use This Identity
Expression has sec or csc	Replace with 1/cos or 1/sin
Expression has tan or cot	Replace with sin/cos or cos/sin
You see sin² + cos²	Replace with 1
You see 1 + tan²	Replace with sec²
Angle is 90° or 180° minus something	Use co-function or supplementary rules
You see a product like sin θ cos θ	Consider sin 2θ = 2 sin θ cos θ
You see (a ± b)²	Expand and look for Pythagorean forms

Final Warning

You can't learn this by reading. Work through at least 20 problems before your exam. Use this guide as a worksheet. The patterns only click when you've struggled through them yourself.

When you get stuck, don't look at the answer immediately. Spend 5 minutes trying every conversion you know. That's how you build the recognition speed you need.