Exam-style CBSE practice questions on Trigonometric Identities (INTRODUCTION TO TRIGONOMETRY). Try each one first, then reveal the correct answer and a step-by-step explanation. Free, from EduLevel — the AI teacher for CBSE.
The expression cos θ / √(1 - cos² θ) is equivalent to which of the following?
Recall the fundamental trigonometric identity that relates sin² θ and cos² θ.
Explanation: Using the identity sin² θ + cos² θ = 1, the denominator √(1 - cos² θ) simplifies to sin θ. The expression becomes cos θ / sin θ, which is equal to cot θ.
What is the value of the expression (tan A ⋅ cosec A)² - (sin A ⋅ sec A)²?
Remember to express all trigonometric functions in terms of sin and cos.
Explanation: The expression simplifies to sec² A - tan² A. Based on the Pythagorean identity 1 + tan² A = sec² A, the value of sec² A - tan² A is 1.
Determine the value of the expression tan² θ - (1/cosθ × secθ).
Recall the fundamental trigonometric identities that relate tangent, secant, and cosine.
Explanation: Since secθ = 1/cosθ, the expression becomes tan² θ - (secθ × secθ) = tan² θ - sec² θ. From the identity sec² θ - tan² θ = 1, it follows that tan² θ - sec² θ = -1.
Which of the following expressions is equivalent to (cotθ + tanθ)?
Try expressing cotangent and tangent in terms of sine and cosine.
Explanation: Expressing in terms of sin and cos, we get (cosθ/sinθ) + (sinθ/cosθ) = (cos²θ + sin²θ)/(sinθcosθ). This simplifies to 1/(sinθcosθ), which is equal to cosecθ secθ.
Identify the equation that holds true for all values of θ where 0° ≤ θ ≤ 90°.
Consider the fundamental trigonometric identities that relate different trigonometric functions.
Explanation: The fundamental Pythagorean identity 1 + tan²θ = sec²θ can be rearranged to sec²θ - tan²θ = 1, which is true for all valid values of θ.
What is the value of the expression (sec² θ - 1)(cosec² θ - 1)?
Recall the fundamental trigonometric identities that relate trigonometric functions to each other.
Explanation: Using the trigonometric identities sec² θ - 1 = tan² θ and cosec² θ - 1 = cot² θ, the expression simplifies to tan² θ * cot² θ, which is equal to 1.
The expression 1 / (cosec θ (1 - cot θ)) + 1 / (sec θ (1 - tan θ)) is equal to which of the following?
Try expressing cosec θ, cot θ, sec θ, and tan θ in terms of sin θ and cos θ.
Explanation: By converting all trigonometric ratios to their sin θ and cos θ forms and simplifying the resulting complex fraction, the expression simplifies to sin θ + cos θ.
If it is given that sin²θ + sinθ = 1, what is the value of the expression cos²θ + cos⁴θ?
Consider the fundamental trigonometric identity relating sine and cosine.
Explanation: From sin²θ + sinθ = 1, we get sinθ = 1 - sin²θ = cos²θ. Substituting this into the expression gives cos²θ + (cos²θ)² = sinθ + sin²θ, which is equal to 1 from the given equation.
What is the distance between the point (a cos θ + b sin θ, 0) and the point (0, a sin θ – b cos θ)?
To find the distance between two points in a coordinate plane, you should recall the distance formula.
Explanation: Using the distance formula, the squared distance is (a cos θ + b sin θ)² + (a sin θ - b cos θ)². Expanding and simplifying using the identity sin²θ + cos²θ = 1 results in a² + b². The distance is the square root, which is √a² + b².
If sec θ + tan θ + 1 = 0, what is the value of sec θ - tan θ?
Recall the fundamental trigonometric identity relating secant and tangent: sec2 θ - tan2 θ = 1.
Explanation: From the given equation, sec θ + tan θ = -1. Using the identity sec²θ - tan²θ = 1, which factors to (sec θ - tan θ)(sec θ + tan θ) = 1, we can substitute the given value: (sec θ - tan θ)(-1) = 1. Therefore, sec θ - tan θ = -1.
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