CBSE Class 10 Maths Trigonometric Ratios & Identities – Quick Guide

Ever wondered why the angles in a right‑angled triangle seem to whisper secret shortcuts? Those whispers are the trigonometric ratios and identities that make CBSE Class 10 maths feel like a magic trick.

In simple words, trigonometric ratios tell you how the sides of a right‑angled triangle compare to each other. Trigonometric identities are equations that let you swap one ratio for another, saving you time on exams.

What are Trigonometric Ratios?

Definition in plain words

A ratio is just a way of saying “how many times bigger one thing is than another.” In a right‑angled triangle, the three main ratios are:

• sine (sin) – opposite side ÷ hypotenuse
• cosine (cos) – adjacent side ÷ hypotenuse
• tangent (tan) – opposite side ÷ adjacent side

Think of water flowing through a pipe: the amount of water (flow) compared to the pipe’s width is like a ratio. Here, the “flow” is a side length, and the “pipe width” is another side.

Key formulas you must remember

For any right‑angled triangle with angle θ (theta):

Memorise the first three – the rest are just their “reciprocals” (flipped versions).

Trigonometric Identities – The Handy Shortcuts

Why we need identities

Imagine you have a big jigsaw puzzle. Identities are the picture on the box that helps you see where each piece belongs, letting you replace a complicated piece with a simpler one.

Basic identities you’ll use often

• Pythagorean identity: sin²θ + cos²θ = 1 (the squares of sine and cosine always add up to 1)
• Reciprocal identities: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
• 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

Proving an identity – a step‑by‑step example

Show that 1 + tan²θ = sec²θ.

1. Start with the Pythagorean identity: sin²θ + cos²θ = 1.
2. Divide every term by cos²θ (the adjacent side squared). You get (sin²θ / cos²θ) + 1 = 1 / cos²θ.
3. Recognise sin²θ / cos²θ as tan²θ (since tan = opposite/adjacent). Also, 1 / cos²θ is sec²θ (the reciprocal of cosine).
4. Rewrite: tan²θ + 1 = sec²θ. Done!

Notice how the identity turned a messy fraction into a clean equation – exactly what exam questions love.

Quick Reference Table

How to Solve Common Class 10 Problems

Example 1: Find a missing side using the sine ratio

Given: a right‑angled triangle where angle θ = 30°, hypotenuse = 10 cm. Find the opposite side.

Step 1: Write the sine formula – sin θ = opposite ÷ hypotenuse.

Step 2: Plug in values – sin 30° = opposite ÷ 10.

Step 3: sin 30° is 0.5 (a standard value you’ll see in the CBSE table). So, 0.5 = opposite ÷ 10.

Step 4: Multiply both sides by 10 → opposite = 0.5 × 10 = 5 cm.

Example 2: Simplify an expression using identities

Simplify (1 – cos θ) / sin θ .

Step 1: Multiply numerator and denominator by (1 + cos θ) – a classic “rationalising” trick.

Step 2: Numerator becomes (1 – cos²θ) which, by the Pythagorean identity, equals sin²θ.

Step 3: Denominator becomes sin θ(1 + cos θ).

Step 4: Cancel one sin θ from numerator and denominator → sin θ / (1 + cos θ).

That’s the simplest form, and you just used two identities in one go.

📝 Likely Exam Questions

1. Find the value of sin 45° without using a calculator.
   Answer: In a 45°‑45°‑90° triangle the legs are equal, so sin 45° = opposite/hypotenuse = √2/2.
2. Prove that tan²θ + 1 = sec²θ.
   Answer: Start from sin²θ + cos²θ = 1, divide by cos²θ, get tan²θ + 1 = sec²θ.
3. If a right‑angled triangle has adjacent side 7 cm and angle θ = 60°, find the hypotenuse.
   Answer: cos θ = adjacent/hypotenuse → cos 60° = 7 / hypotenuse → 0.5 = 7 / h → h = 14 cm.
4. Simplify (sec θ – tan θ)(sec θ + tan θ).
   Answer: It is a difference of squares → sec²θ – tan²θ. Using identity sec²θ = 1 + tan²θ, the expression becomes 1.
5. Express sin θ in terms of tan θ.
   Answer: tan θ = sin θ / cos θ → sin θ = tan θ·cos θ. From sin²θ + cos²θ = 1, cos θ = 1/√(1+tan²θ). Hence sin θ = tan θ / √(1+tan²θ).