Trigonometric Identities Questions for CDS

Fundamental Ratios and Reciprocal Identities

There are six trig ratios. Three are primary — sin, cos, tan — and three are their reciprocals. Every complex identity eventually breaks down into these six. If you forget any formula in the exam, convert everything to sin and cos first — that is your emergency reset button.

• sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ / cos θ
• cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ = cos θ / sin θ
• tan θ × cot θ = 1 always
• sin θ × cosec θ = 1, cos θ × sec θ = 1
• Converting to sin/cos is the universal fallback strategy

Pythagorean Identities

These three identities are the most heavily tested in CDS. They come directly from the Pythagoras theorem applied to a unit circle. Think of them as a family of three siblings — once you know the parent (sin²θ + cos²θ = 1), the other two are just rearrangements using reciprocals. Examiners disguise these by writing them in rearranged or factored forms.

• sin²θ + cos²θ = 1 (the parent identity)
• 1 + tan²θ = sec²θ (divide parent by cos²θ)
• 1 + cot²θ = cosec²θ (divide parent by sin²θ)
• Rearranged forms tested: sin²θ = 1 - cos²θ, sec²θ - tan²θ = 1, cosec²θ - cot²θ = 1
• Factored form often tested: (sec θ - tan θ)(sec θ + tan θ) = 1

Complementary Angle Identities

These identities involve angles that add up to 90 degrees. The rule is simple: the co-function of an angle equals the function of its complement. The prefix co in cosine, cotangent, cosecant is a clue — co stands for complement. CDS often tests these in simplification questions where sin 70 appears alongside cos 20 and you need to see they are equal.

• 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
• Trick: anything with (90 - θ) flips to its co-partner
• Common exam trap: sin 35 / cos 55 = sin 35 / sin 35 = 1 since cos 55 = sin 35

Sum and Difference Formulas

These expand trig of a sum or difference into products. CDS tests these mostly in find-the-value questions where you split a non-standard angle like 75 degrees into 45 + 30. Memorise the sign pattern: for sin the signs match (plus gives plus, minus gives minus), for cos the signs flip.

• sin(A + B) = sinA cosB + cosA sinB
• sin(A - B) = sinA cosB - cosA sinB
• cos(A + B) = cosA cosB - sinA sinB (sign flips)
• cos(A - B) = cosA cosB + sinA sinB (sign flips)
• tan(A + B) = (tanA + tanB) / (1 - tanA tanB)
• tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

Double Angle and Half Angle Formulas

Double angle formulas are a special case of sum formulas where A = B. CDS tests these in simplification questions and in questions that give you sin θ or cos θ and ask for sin 2θ. The cos 2θ identity has three equivalent forms — choose the one that matches what is given in the question.

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
• tan 2θ = 2 tan θ / (1 - tan²θ)
• Half angle: sin²(θ/2) = (1 - cos θ)/2, cos²(θ/2) = (1 + cos θ)/2
• Choose the cos 2θ form based on what the question gives you — only sin given: use 1 - 2sin²θ, only cos given: use 2cos²θ - 1

Standard Angle Values Table

These values must be memorised without hesitation. A useful pattern for sin: sin 0 = √0/2, sin 30 = √1/2, sin 45 = √2/2, sin 60 = √3/2, sin 90 = √4/2. For cos, the order reverses. Tan is sin/cos.

• sin 0=0, sin 30=1/2, sin 45=1/√2, sin 60=√3/2, sin 90=1
• cos 0=1, cos 30=√3/2, cos 45=1/√2, cos 60=1/2, cos 90=0
• tan 0=0, tan 30=1/√3, tan 45=1, tan 60=√3, tan 90=undefined
• sin 90=1, sin 180=0, sin 270=-1, sin 360=0
• cos 180=-1, cos 270=0, cos 360=1

• Writing cos(A+B) = cosA + cosB — trig does NOT distribute over addition. This is the most common error.
• Confusing the sign in cos(A+B) vs cos(A-B). Remember: cos flips the sign (cos sum has minus, cos difference has plus).
• Using the wrong form of cos 2θ. If only sin θ is given use 1 - 2sin²θ, not cos²θ - sin²θ which needs cos θ too.
• Forgetting that sec²θ - tan²θ = 1, not tan²θ - sec²θ = 1. Sign order matters in the Pythagorean reciprocal forms.
• In complementary angles, computing cos 55 separately instead of recognising cos 55 = sin 35 instantly and cancelling.

• Pythagoras Family: S+C=1, then divide by C² to get T+1=S(ec)², divide by S² to get Co+1=Co(sec)². The parent spawns two children by division.
• CAST Rule for signs in four quadrants: All Students Take Coffee — All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4.
• Co-function shortcut: CO in the name = COmplement partner. cosine pairs with sine, cotangent pairs with tangent, cosecant pairs with secant.
• Sin double angle: Two Sin Cos (2sinθcosθ) — just think Two S C.
• Standard sin values pattern: 0, 1/2, 1/√2, √3/2, 1 — count the numerators as √0, √1, √2, √3, √4 all over 2.

• CDS typically places 3-5 trig identity questions in the paper. They are usually faster than geometry or mensuration — target them early to bank time.
• Proving identities: always start from the more complicated side and simplify. Never cross-multiply across the equals sign when proving.
• If a question gives sin θ + cos θ = k, squaring both sides is almost always the key move — it introduces the Pythagorean identity and the product sin θ cos θ.
• Questions like find value of (sin θ + cosec θ)² + (cos θ + sec θ)² often expand to a clean integer — expand fully, apply identities, expect answer like 7 or 9.
• Watch for expression sin A cos B type questions — these often resolve using sum-to-product or product-to-sum shortcuts. If you see a product of two trig terms, suspect a double angle or sum formula is the intended route.