Notes Formulae Read NCERT Chapter 8 (jemh108) Practice Check my answer NCERT Solutions 16 NCERT Exemplar questions from Chapter 8
Exercise 8.2 Exemplar Ch8 Ex 8.2 Q4 2 marks State whether the statement is true or false, with justification: (1 − cos²θ) · sec²θ = tan θ.
Stuck? Show a hint Step 1 LHS = (1 − cos²θ) · sec²θ = sin²θ · (1/cos²θ) = sin²θ/cos²θ = tan²θ.
Exercise 8.3 Exemplar Ch8 Ex 8.3 Q1 3 marks Prove that sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2 cosec θ.
Stuck? Show a hint Step 1 LHS = sin θ/(1 + cos θ) + (1 + cos θ)/sin θ.
Step-by-step solution LHS = sin θ/(1 + cos θ) + (1 + cos θ)/sin θ. Combine over common denominator sin θ · (1 + cos θ): LHS = [sin²θ + (1 + cos θ)²] / [sin θ (1 + cos θ)]. Expand numerator: sin²θ + 1 + 2 cos θ + cos²θ = (sin²θ + cos²θ) + 1 + 2 cos θ = 1 + 1 + 2 cos θ = 2 + 2 cos θ = 2(1 + cos θ). LHS = 2(1 + cos θ) / [sin θ (1 + cos θ)] = 2/sin θ = 2 cosec θ = RHS. Hence proved. Answer: LHS = 2 cosec θ = RHS.
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Exemplar Ch8 Ex 8.3 Q3 3 marks If tan A = 3/4, show that sin A · cos A = 12/25.
Stuck? Show a hint Step 1 tan A = 3/4 ⇒ opposite = 3k, adjacent = 4k. Hypotenuse = √((3k)² + (4k)²) = √(25k²) = 5k.
Exemplar Ch8 Ex 8.3 Q9 2 marks If √3 tan θ = 1, find the value of sin²θ − cos²θ.
Stuck? Show a hint Step 1 tan θ = 1/√3 ⇒ θ = 30°.
Exemplar Ch8 Ex 8.3 Q11 3 marks Simplify (1 + tan²θ)(1 − sin θ)(1 + sin θ).
Stuck? Show a hint Step 1 (1 − sin θ)(1 + sin θ) = 1 − sin²θ = cos²θ.
Exemplar Ch8 Ex 8.3 Q12 2 marks If 2 sin²θ − cos²θ = 2, then find the value of θ (0° ≤ θ ≤ 90°).
Stuck? Show a hint Step 1 Replace cos²θ with 1 − sin²θ: 2 sin²θ − (1 − sin²θ) = 2.
Exercise 8.4 Exemplar Ch8 Ex 8.4 Q1 5 marks If cosec θ + cot θ = p, prove that cos θ = (p² − 1)/(p² + 1).
Stuck? Show a hint Step 1 Given: cosec θ + cot θ = p. …(1)
Step-by-step solution Given: cosec θ + cot θ = p. …(1) Use the identity cosec²θ − cot²θ = 1, i.e., (cosec θ + cot θ)(cosec θ − cot θ) = 1. So cosec θ − cot θ = 1/p. …(2) Add (1) and (2): 2 cosec θ = p + 1/p = (p² + 1)/p ⇒ cosec θ = (p² + 1)/(2p) ⇒ sin θ = 2p/(p² + 1). Subtract (2) from (1): 2 cot θ = p − 1/p = (p² − 1)/p ⇒ cot θ = (p² − 1)/(2p). cos θ = cot θ · sin θ = [(p² − 1)/(2p)] · [2p/(p² + 1)] = (p² − 1)/(p² + 1). Proved. Answer: cos θ = (p² − 1)/(p² + 1).
See exactly which step lost you marks: photograph your working (first check free)
Exemplar Ch8 Ex 8.4 Q9 5 marks If tan θ + sec θ = l, prove that sec θ = (l² + 1)/(2l).
Stuck? Show a hint Step 1 [1 mark] Given: tan θ + sec θ = l. …(1)
Step-by-step solution [1 mark] Given: tan θ + sec θ = l. …(1) [1 mark] Use the identity sec²θ − tan²θ = 1, i.e., (sec θ + tan θ)(sec θ − tan θ) = 1. [1 mark] Dividing by (1): sec θ − tan θ = 1/l. …(2) [1 mark] Add (1) and (2): 2 sec θ = l + 1/l = (l² + 1)/l. [1 mark] Therefore sec θ = (l² + 1)/(2l). Proved. Answer: sec θ = (l² + 1)/(2l).
See exactly which step lost you marks: photograph your working (first check free)
Examples and more questions Exemplar Ch8 MCQ Q1 1 mark If cos A = 4/5, then the value of tan A is:
(a) 3/5 (b) 3/4 (c) 4/3 (d) 5/3 Stuck? Show a hint Step 1 cos A = adjacent/hypotenuse = 4/5: let adjacent = 4k, hypotenuse = 5k.
Exemplar Ch8 MCQ Q2 1 mark If sin A = 1/2, then the value of cot A is:
(a) √3 (b) 1/√3 (c) √3/2 (d) 1 Stuck? Show a hint Step 1 sin A = 1/2 ⇒ A = 30°.
Exemplar Ch8 MCQ Q4 1 mark Given that sin θ = a/b, then cos θ is equal to:
(a) b/√(b² − a²) (b) b/a (c) √(b² − a²)/b (d) a/√(b² − a²) Stuck? Show a hint Step 1 sin²θ + cos²θ = 1 ⇒ cos²θ = 1 − (a/b)² = (b² − a²)/b².
Exemplar Ch8 MCQ Q8 1 mark If ∆ABC is right-angled at C, then the value of cos(A + B) is:
(a) 0 (b) 1 (c) 1/2 (d) √3/2 Stuck? Show a hint Step 1 Sum of angles in a triangle = 180°. Since C = 90°, A + B = 90°.
Exemplar Ch8 MCQ Q9 1 mark If sin A + sin²A = 1, then the value of the expression (cos²A + cos⁴A) is:
(a) 1 (b) 1/2 (c) 2 (d) 3 Stuck? Show a hint Step 1 Given sin A + sin²A = 1 ⇒ sin A = 1 − sin²A = cos²A.
Exemplar Ch8 MCQ Q10 1 mark Given that sin α = 1/2 and cos β = 1/2, then the value of (α + β) is:
(a) 0° (b) 30° (c) 60° (d) 90° Stuck? Show a hint Step 1 sin α = 1/2 ⇒ α = 30°.
Exemplar Ch8 MCQ Q12 1 mark If 4 tan θ = 3, then (4 sin θ − cos θ)/(4 sin θ + cos θ) is equal to:
(a) 2/3 (b) 1/3 (c) 1/2 (d) 3/4 Stuck? Show a hint Step 1 Divide numerator and denominator by cos θ: numerator → 4 tan θ − 1; denominator → 4 tan θ + 1.
Exemplar Ch8 MCQ Q13 1 mark If sin θ − cos θ = 0, then the value of (sin⁴θ + cos⁴θ) is:
(a) 1 (b) 3/4 (c) 1/2 (d) 1/4 Stuck? Show a hint Step 1 sin θ = cos θ ⇒ tan θ = 1 ⇒ θ = 45°.
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