Class 10 Trigonometric Identities — practice questions with answers

Trigonometry · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.

1. Which of the following is the fundamental Pythagorean identity?

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Answer: A) sin² A + cos² A = 1

By definition, for any angle A, sin² A + cos² A = 1. This is the basic trigonometric identity.

2. Simplify: cos² A - sin² A + 1.

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Answer: A) 2 cos² A

Replace 1 with sin² A + cos² A: cos² A - sin² A + sin² A + cos² A = 2 cos² A.

3. Simplify:

\frac{s i n A}{cosec A} + \frac{c o s A}{s e c A}

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Answer: D) 1

sin A/cosec A = sin A * sin A = sin² A. cos A/sec A = cos A * cos A = cos² A. Sum = sin² A + cos² A = 1.

4. For an acute angle A,

\frac{1 + s i n A}{1 - s i n A}

equals which expression?

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Answer: A)

sec A + tan A

Multiply numerator and denominator inside the root by

1 + sin A

\frac{( 1 + s i n A ) ^{2}}{1 - s i n ^{2} A} = \frac{( 1 + s i n A ) ^{2}}{c o s ^{2} A} = \frac{1 + s i n A}{c o s A} = sec A + tan A

, positive because A is acute. Multiplying by

1 - sin A

instead gives

sec A - tan A

, which is the RECIPROCAL of the required value, not the value.

5. A designer models a point on a circular running track of radius 5 m as

x = 5 cos θ

and

y = 5 sin θ

, with

θ

measured at the centre.
Eliminating

θ

, which equation do x and y always satisfy?

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Answer: C)

x^{2} + y^{2} = 25

Squaring and adding:

x^{2} + y^{2} = 25 cos^{2} θ + 25 sin^{2} θ = 25 (cos^{2} θ + sin^{2} θ) = 25

. The Pythagorean identity does the whole job. Answering

x^{2} + y^{2} = 5

forgets to square the radius as well; answering

x + y = 5

adds the two equations instead of squaring them first.

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