Trigonometric Identities for SSC

SSC trigonometry rarely asks you to find an angle. Instead, it asks you to simplify an expression or find one ratio from another, and the tools are a short list of identities. The most used are the three Pythagorean identities, all from the right triangle with hypotenuse 1:

Identity	Rearranged forms
sin²θ + cos²θ = 1	1 − sin²θ = cos²θ; 1 − cos²θ = sin²θ
1 + tan²θ = sec²θ	sec²θ − tan²θ = 1
1 + cot²θ = cosec²θ	cosec²θ − cot²θ = 1

The second and third come from dividing the first by cos²θ and by sin²θ. You only need to learn one and derive the others.

Worked example 1: the first identity

sin² 30° + cos² 30° = 1/4 + 3/4 = 1.

Worked example 2: the second identity

sec² 60° − tan² 60° = 4 − 3 = 1.

When two angles add to 90°, each ratio of one equals the co-ratio of the other:

sin(90° − θ) = cos θ and cos(90° − θ) = sin θ
tan(90° − θ) = cot θ and cot(90° − θ) = tan θ
sec(90° − θ) = cosec θ and cosec(90° − θ) = sec θ

Worked example 3: sin 37° ÷ cos 53°

37° + 53° = 90°, so sin 37° = cos 53° and the ratio is 1.

Products of tangents use the fact that tan θ × tan(90° − θ) = tan θ × cot θ = 1.

Worked example 4: a product of tangents

tan 10° × tan 20° × tan 70° × tan 80° pairs into (tan 10° × tan 80°)(tan 20° × tan 70°) = 1 × 1 = 1.

The identity sec²θ − tan²θ = 1 factorises as (sec θ + tan θ)(sec θ − tan θ) = 1. So if one bracket is k, the other is 1/k. Adding and subtracting the two then gives sec θ and tan θ separately. The same works for cosec θ ± cot θ.

Worked example 5: sec θ + tan θ = 3

Then sec θ − tan θ = 1/3. Adding: 2 sec θ = 10/3, so sec θ = 5/3, and tan θ = 3 − 5/3 = 4/3.

For any a and b, the expression a sin θ + b cos θ ranges from −√(a² + b²) to +√(a² + b²). Questions ask for the maximum, the minimum, or both.

Worked example 6: 3 sin θ + 4 cos θ

The maximum is √(9 + 16) = 5, and the minimum is −5. It is not 7, because sin θ and cos θ cannot both equal 1 at the same angle.

Other common ranges: sin θ and cos θ lie between −1 and 1, while sec θ and cosec θ are never between −1 and 1. And sin²θ + cosec²θ is always at least 2, with equality at θ = 90°.

If a question says "simplify" and the options are expressions, the answer is an identity, and an identity must hold for every allowed angle. Put a convenient angle into the question and into each option; the option that matches is correct. Use 45° for expressions with tan and cot, and 30° or 60° when 45° makes too many terms equal.

Worked example 7: (1 − sin²θ) ÷ cos²θ

At θ = 45°: (1 − 1/2) ÷ (1/2) = 1. Using the identity directly, 1 − sin²θ = cos²θ, so the expression is always 1.

Worked example 8: sin⁴θ − cos⁴θ

Factorise as (sin²θ + cos²θ)(sin²θ − cos²θ) = sin²θ − cos²θ. At θ = 60°, both sides give 9/16 − 1/16 = 1/2 and 3/4 − 1/4 = 1/2, a difference of 0, which confirms the identity.

✘ Writing sin²θ − cos²θ = 1. The identity is a sum.
✘ Adding coefficients for a maximum: the maximum of a sin θ + b cos θ is √(a² + b²).
✘ Choosing a value-putting angle that makes a denominator zero.
✘ Confusing sin²θ with sin θ².

Standard values of the ratios are the inputs for every method here; questions on angles and shapes also appear in polygon angles. The algebraic identities used to factorise, and the formula list in the Quant cheat code, complete the toolkit.

Q1. What is the value of 1 + tan² 45°? easy

(a) 1 (b) 2 (c) 0 (d) √2

✅ Answer: (b) 2

💡 Explanation: 1 + tan²θ = sec²θ. With tan 45° = 1, the value is 1 + 1 = 2, which also equals sec² 45° = (√2)².

Q2. What is the value of sin 25° ÷ cos 65°? easy

(a) 1 (b) 0 (c) tan 25° (d) 2

✅ Answer: (a) 1

💡 Explanation: 25° and 65° add to 90°, so sin 25° = cos(90° − 25°) = cos 65°. The ratio is 1.

Q3. What is tan 1° × tan 2° × tan 3° × … × tan 89°? medium

(a) 0 (b) 89 (c) 1 (d) √3

✅ Answer: (c) 1

💡 Explanation: Pair tan θ with tan(90° − θ) = cot θ: each pair multiplies to 1. The middle term is tan 45° = 1, so the whole product is 1.

Q4. If cosec θ − cot θ = 1/4, what is cosec θ + cot θ? medium

(a) 1/4 (b) 3/4 (c) 2 (d) 4

✅ Answer: (d) 4

💡 Explanation: cosec²θ − cot²θ = 1 factorises as (cosec θ − cot θ)(cosec θ + cot θ) = 1. So cosec θ + cot θ = 1 ÷ (1/4) = 4.

Q5. What is the maximum value of 5 sin θ + 12 cos θ? medium

(a) 17 (b) 12 (c) 13 (d) 7

✅ Answer: (c) 13

💡 Explanation: The maximum of a sin θ + b cos θ is √(a² + b²) = √(25 + 144) = √169 = 13. Adding 5 and 12 is wrong because sin and cos never both reach 1 at once.

Q6. What is the value of sec² 45° + cosec² 45°? easy

(a) 4 (b) 2 (c) 1 (d) 8

✅ Answer: (a) 4

💡 Explanation: sec 45° = cosec 45° = √2, so each square is 2 and the sum is 4. In general sec²θ + cosec²θ = sec²θ × cosec²θ.

What is the value-putting method?

If an expression is an identity, it gives the same result for every allowed angle. Substitute a convenient angle such as 45°, 30° or 0°, evaluate each option the same way, and pick the one that matches. Avoid angles that make a denominator zero.

Why does the maximum of a sin θ + b cos θ equal √(a² + b²)?

The expression can be rewritten as √(a² + b²) × sin(θ + α) for a fixed angle α. Since sine never exceeds 1, the largest value is √(a² + b²), and the smallest is its negative.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 14 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-23 against the sources listed above.

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