Trigonometric identities for NET: formulas, shortcuts and worked MCQs

Trigonometric identities on NET test the fundamental identities, the sum and difference formulas, double and half angles, and sum-to-product conversions. You need around 25 formulas memorised exactly. With them, most questions collapse to one line, and without them no amount of algebra saves the clock.

Key facts

• Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
• Trigonometric Identities is unit 10 of the Class 11 PECTAA mathematics book (National Curriculum 2023)
• Our chapter map estimates about 9 MCQs from this chapter, among the highest of any maths chapter. NUST does not publish chapter counts

Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map

What the chapter covers#

The chapter starts from sin^2 θ + cos^2 θ = 1 and builds out: the other Pythagorean identities, allied angles, the sum and difference formulas, double and half angles, and conversions between sums and products. It ends with proving identities, which on NET means simplifying an expression until it matches an option.

This chapter has the largest practice set of any maths chapter on the site. Our bank for it also covers triangle solving with the sine and cosine rules, heights and distances, and some graph and equation questions from the neighbouring trigonometry units. Expect trigonometry to be spread across the paper, not boxed into one chapter.

Formulas#

Fundamental:

sin^2 θ + cos^2 θ = 1
1 + tan^2 θ = sec^2 θ
1 + cot^2 θ = csc^2 θ

Sum and difference:

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Double, triple and half angle:

sin 2A = 2 sin A cos A
cos 2A = cos^2 A - sin^2 A = 1 - 2 sin^2 A = 2 cos^2 A - 1
tan 2A = 2 tan A / (1 - tan^2 A)
sin 3A = 3 sin A - 4 sin^3 A
cos 3A = 4 cos^3 A - 3 cos A
sin^2(A/2) = (1 - cos A)/2
cos^2(A/2) = (1 + cos A)/2
tan(A/2)   = sin A / (1 + cos A) = (1 - cos A) / sin A

Sum to product and product to sum:

sin C + sin D = 2 sin((C + D)/2) cos((C - D)/2)
sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2)
cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2)
cos C - cos D = -2 sin((C + D)/2) sin((C - D)/2)

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

Allied angles: subtract multiples of 360° until the angle lies between 0° and 360°. Then use the quadrant rule for the sign. In quadrant I all ratios are positive, in II only sin and csc, in III only tan and cot, and in IV only cos and sec.

Worked MCQs#

Five questions of the type the practice bank uses most. We checked each one with a test angle.

1. If sec θ - tan θ = x, find sec θ + tan θ.

(sec θ - tan θ)(sec θ + tan θ) = sec^2 θ - tan^2 θ = 1
sec θ + tan θ = 1/x

2. Find cos^2(45° + x) - sin^2(45° - x).

The two angles add to 90°, so sin(45° - x) = cos(45° + x). The two terms are equal and the answer is 0.

3. Write cos(3α/4) + cos(4α/3) as a product.

(C + D)/2 = (9α/12 + 16α/12)/2 = 25α/24
(C - D)/2 = (9α/12 - 16α/12)/2 = -7α/24
cos C + cos D = 2 cos(25α/24) cos(7α/24)

Cosine is even, so the minus sign on 7α/24 drops out.

4. If cos A / 3 = cos B / 4 = 1/5, with A and B between -90° and 0°, find 2 sin A + 4 sin B.

cos A = 3/5,   sin A = -4/5      (fourth quadrant, sine negative)
cos B = 4/5,   sin B = -3/5
2(-4/5) + 4(-3/5) = -8/5 - 12/5 = -4

The option +4 is waiting for anyone who ignores the quadrant.

5. What is the sign of tan 2140°?

2140° - 5(360°) = 340°

340° is in quadrant IV, where tan is negative.

Common traps#

• Sign in cos(A + B). The sign inside the formula is the opposite of the sign in the angle. cos(A + B) has a minus.
• Squaring away the quadrant. sin A = ±sqrt(1 - cos^2 A). The question's range for A decides the sign, as in example 4.
• Complementary pairs. tan 40° = cot 50°, sin 20° = cos 70°. Questions like tan 40°/cot 50° + cot 50°/tan 40° are built to collapse to 1 + 1 = 2.
• Three forms of cos 2A. Pick the form that matches the rest of the expression. If the question has only sines, use 1 - 2 sin^2 A.
• Degrees vs radians. cos(-15π) is cos 15π = cos π = -1. Students treat 15π as degrees.

How to prepare#

Write the formula blocks above from memory every day for a week. That sounds dull, and it is the only reliable way to get them exact. Then do timed sets in small batches. With the size of the practice bank for this chapter, you will not run out of questions.

Complex numbers uses these identities through polar form, and differentiation and integration both assume you can rewrite sin^2 x or cos 2x without thinking.

FAQ#