Basic Identities and Operations — JEE Main Mathematics MCQs with Solutions
Free JEE Main Mathematics Basic Identities and Operations MCQs with step-by-step solutions (33 questions). Part of Trigonometric Ratios. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Basic Identities and Operations · easy
sec A + tan A = 3 ⇒ sec A =
A. $\frac{10}{3}$
B. $\frac{5}{3}$ ✓ Correct
C. $\frac{2}{3}$
D. $\frac{4}{3}$
Solution: Using $\sec^2 A - \tan^2 A = 1$ and the given relation.
Q2 — Basic Identities and Operations · easy
sec A – tan A = 4 ⇒ tan A =
A. $\frac{15}{8}$
B. $-\frac{15}{8}$ ✓ Correct
C. $\frac{17}{8}$
D. $-\frac{17}{8}$
Solution: Apply the identity $(\sec A + \tan A)(\sec A - \tan A) = 1$.
Q3 — Basic Identities and Operations · easy
sec A – tan A = 5 ⇒ sin A =
A. $\frac{6}{13}$
B. $-\frac{6}{13}$
C. $\frac{12}{13}$ ✓ Correct
D. $-\frac{12}{13}$
Solution: From $\sec A - \tan A = 5$, find $\sec A$ then $\sin A$.
Q4 — Basic Identities and Operations · easy
cosec A – cot A = 5 ⇒ cosec A =
A. $\frac{5}{13}$
B. $\frac{13}{5}$ ✓ Correct
C. $-\frac{5}{13}$
D. $-\frac{13}{5}$
Solution: Use $(\cosec A - \cot A)(\cosec A + \cot A) = 1$.
Q5 — Basic Identities and Operations · easy
cosec A – cot A = 6 ⇒ cot A =
A. $\frac{12}{35}$
B. $-\frac{12}{35}$ ✓ Correct
C. $\frac{35}{12}$
D. $-\frac{35}{12}$
Solution: Solve using $\cosec^2 A - \cot^2 A = 1$.
Q6 — Basic Identities and Operations · medium
cosec A + cot A = $\frac{2}{3}$ ⇒ cos A =
A. $\frac{5}{13}$
B. $\frac{13}{5}$
C. $-\frac{5}{13}$ ✓ Correct
D. $-\frac{13}{5}$
Solution: From the sum and the identity, derive cos A.
Q7 — Basic Identities and Operations · medium
sec θ – tan θ = 3 ⇒ θ lies in the quadrant
A. I
B. II
C. III
D. IV ✓ Correct
Solution: Since $\sec θ - \tan θ = 3 > 0$, analyze sign conditions.
Q8 — Basic Identities and Operations · medium
cosec θ – cot θ = 5 ⇒ θ lies in the quadrant
A. I ✓ Correct
B. II
C. III
D. IV
Solution: $\cosec θ > \cot θ$ in quadrant I where both are positive.
Q9 — Basic Identities and Operations · medium
sin θ = $\frac{2ab}{a^2 + b^2}$ ⇒ sec θ + tan θ =
A. $\frac{a - b}{a + b}$
B. $\frac{a + b}{a - b}$ ✓ Correct
C. $\frac{ab}{a^2 + b^2}$
D. $\frac{ab}{a + b}$
Solution: From the given sin, compute cos and sec + tan.
Q10 — Basic Identities and Operations · medium
sin α = $\frac{2xy}{x^2 + y^2}$ ⇒ sec α – tan α =
A. $\frac{x - y}{x + y}$ ✓ Correct
B. $\frac{x + y}{x - y}$
C. $\frac{xy}{x^2 + y^2}$
D. $\frac{xy}{x + y}$
Solution: Calculate from the given sin α using identities.
Q11 — Basic Identities and Operations · medium
sin x = $\frac{2pq}{p^2 + q^2}$ ⇒ cosec x + cot x =
A. $\frac{p}{q}$ ✓ Correct
B. $\frac{q}{p}$
C. $\frac{2p}{q}$
D. $\frac{2q}{p}$
Solution: Derive from sin x using trigonometric identities.
Q12 — Basic Identities and Operations · medium
sin β = $\frac{2mn}{m^2 + n^2}$ ⇒ cosec β – cot β =
A. $\frac{m}{n}$ ✓ Correct
B. $\frac{n}{m}$
C. $\frac{2m}{n}$
D. $\frac{2n}{m}$
Solution: Calculate using the sine value and identities.
Q13 — Basic Identities and Operations · hard
$3[\sin x - \cos x]^4 + 6[\sin x + \cos x]^2 + 4[\sin^6 x + \cos^6 x]$ =
A. 3
B. 6
C. 4
D. 13 ✓ Correct
Solution: Expand and simplify using algebraic identities.
Q14 — Basic Identities and Operations · hard
$\frac{\cos^3 A - \sin^3 A}{\cos A - \sin A} + \frac{\cos^3 A + \sin^3 A}{\cos A + \sin A} = k$, then k =
A. 0
B. 1
C. 2 ✓ Correct
D. –1
Solution: Factor and simplify the algebraic expressions.
Q15 — Basic Identities and Operations · hard
$\frac{\sin^2 α + \tan^2 α \cos α}{1 + \cot^2 α(1 + \tan^2 α)}$ =
A. –1
B. 0
C. 1 ✓ Correct
D. 2
Solution: Simplify using Pythagorean identities.
Q16 — Basic Identities and Operations · hard
$\frac{1}{(1 + \cot^2 α)^2} + \frac{\tan^2 α}{(1 + \tan^2 α)^2} + \frac{1}{1 + \tan^2 α}$ =
A. –1
B. 0
C. 1 ✓ Correct
D. 2
Solution: Use $1 + \cot^2 α = \cosec^2 α$ and $1 + \tan^2 α = \sec^2 α$.
Q17 — Basic Identities and Operations · hard
$\frac{(\sqrt{3} + 2\cos α)^3 + (1 + 2\sin α)^3}{(1 - 2\sin α)^3} + \frac{(\sqrt{3} - 2\cos α)^3}{(\sqrt{3} - 2\sin α)^3}$ =
A. 1
B. $\sqrt{3}$
C. 0 ✓ Correct
D. –1
Solution: Expand and cancel terms carefully.
Q18 — Basic Identities and Operations · hard
$\frac{\cot θ + \cosec θ - 1}{\cot θ - \cosec θ + 1}$ I. (1) $\frac{1 - \cos θ}{\sin θ}$ II. (2) $\frac{\sin θ}{1 - \cos θ}$
A. $\frac{1 - \cos θ}{\sin θ}$ and $\frac{\sin θ}{1 - \cos θ}$ ✓ Correct
B. Other
C. Other
D. Other
Solution: Rationalize and simplify the expression.
Q19 — Basic Identities and Operations · hard
$\frac{1 + \cot θ + \cosec θ}{1 - \cot θ + \cosec θ}$ I. (1) $\frac{\sin α}{1 + \cos α}$ II. (1) $\frac{1 + \cos α}{\sin α}$
A. Type I and II ✓ Correct
B. Other
C. Other
D. Other
Solution: Simplify each part separately.
Q20 — Basic Identities and Operations · hard
$\frac{\tan α + \sec α - 1}{\tan α - \sec α + 1}$ I. (1) $\frac{1 - \sin α}{\cos α}$ II. (1) $\frac{\cos α}{1 - \sin α}$ III. (1) $\sec α + \tan α$
A. All three ✓ Correct
B. Other
C. Other
D. Other
Solution: All three forms are equivalent.
Q21 — Basic Identities and Operations · hard
$\frac{1 + \tan α + \sec α}{1 - \tan α + \sec α}$ I. (1) $\frac{1 - \sin α}{\cos α}$ II. (1) $\frac{\cos α}{1 - \sin α}$
A. Type I and II ✓ Correct
B. Other
C. Other
D. Other
Solution: Work through each simplification.
Q22 — Basic Identities and Operations · hard
$(\sin α + \cosec α)^2 + (\sec α + \cos α)^2 = k + \tan^2 α + \cot^2 α$, k =
A. 9
B. 7 ✓ Correct
C. 5
D. 3
Solution: Expand and collect like terms.
Q23 — Basic Identities and Operations · hard
$\tan^2 θ - \sin^2 θ - \tan^2 θ \sin^2 θ$ =
A. 1
B. 0 ✓ Correct
C. 2
D. –1
Solution: Factor: $\tan^2 θ(1 - \sin^2 θ) - \sin^2 θ = \tan^2 θ \cos^2 θ - \sin^2 θ = \sin^2 θ - \sin^2 θ$.
Q24 — Basic Identities and Operations · hard
If θ is acute, $\tan θ = \frac{1}{\sqrt{7}} - \frac{\cosec^2 θ - \sec^2 θ}{\cosec^2 θ + \sec^2 θ}$ =
A. $\frac{3}{4}$ ✓ Correct
B. $\frac{1}{2}$
C. 2
D. $\frac{5}{4}$
Solution: Simplify the fraction using definitions.
Q25 — Basic Identities and Operations · hard
$\frac{1}{\cosec A - 4p} \implies \cosec A + \cot A$ =
A. $8p$
B. $\frac{1}{8p}$
C. $-8p$ or $\frac{1}{8p}$
D. $\frac{1}{8p}$ or $8p$ ✓ Correct
Solution: Use $(\cosec A - \cot A)(\cosec A + \cot A) = 1$.
Q26 — Basic Identities and Operations · hard
$\frac{1}{\cosec x - \frac{1}{4x}} \implies \cosec β - \cot β$ =
A. $2x$ or $-\frac{1}{2x}$
B. $-2x$ or $\frac{1}{2x}$
C. $-2x$ or $-\frac{1}{2x}$
D. $\frac{1}{2x}$ or $2x$ ✓ Correct
Solution: Apply identity and solve.
Q27 — Basic Identities and Operations · hard
$\frac{1}{\sec 5x - \frac{1}{20x}} \implies \sec α + \tan α$ =
A. $5x$
B. $\frac{1}{20x}$
C. $10x$ or $-\frac{1}{10x}$
D. $10x$ or $\frac{1}{10x}$ ✓ Correct
Solution: From $\sec 5x - \tan 5x = \frac{1}{20x}$.
Q28 — Basic Identities and Operations · hard
$\tan P - \frac{1}{4p} \implies \sec θ - \tan θ$ =
A. $2p$ or $\frac{1}{2p}$
B. $\frac{1}{2p}$ or $-2p$
C. $-\frac{1}{2p}$ or $2p$
D. $-\frac{1}{2p}$ or $-2p$ ✓ Correct
Solution: Use $\sec^2 θ - \tan^2 θ = 1$.
Q29 — Basic Identities and Operations · hard
$\sqrt{\frac{1 - \sin A}{1 + \sin A}}$ =
A. ±(sec A – tan A) ✓ Correct
B. ±(sec A + tan A)
C. ±(cosec A – cot A)
D. ±(cosec A + cot A)
Solution: Rationalize and simplify.
Q30 — Basic Identities and Operations · hard
$\frac{\sqrt{1 + \sin A}}{\sqrt{1 - \sin A}} = k(\sec A + \tan A) \Rightarrow k$ =
A. –1
B. 1
C. ±1 ✓ Correct
D. ±2
Solution: Square to find k.