Class 10 Maths Chapter 8 Question Answers - Introduction to Trigonometry

Q1: In the adjoining figure, ABCD is a rectanlge with breadth BC = 7 cm and ∠CAB = 30°. Find the length of side AB of the rectangle and length of diagonal AC. If the ∠CAB = 60°, then what is the size of the side AB of the rectangle. [Use √3 = 1.73 and √2 = 1.41, if required)
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Q2: Prove that:

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Q3: If x = r sin A cos C, y = r sin A sin C and z = r cos A, then prove that x² + y² + z² = r². 
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x = r sin A cos C; y = r sin A sin C; z = r cos A
Squaring and adding,
L.H.S. x² + y² + z² = 2 sin² A cos²C + r² sin² A sin² C + r² cos² A
= r² sin² A(cos² C + sin² C) + r² cos² A
= r² sin² A + r² cos² A … [cos²θ + sin²θ = 1
= r² (sin² A + cos² A) = r² = R.H.S.

Q4: Prove that: (cot A + sec B)² – (tan B – cosec A)² = 2(cot A . sec B + tan B. cosec A)
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L.H.S.
= (cot A + sec B)² – (tan B – cosec A)²
= cot² A + sec² B + 2 cot A sec B – (tan² B + cosec² A – 2 tan B cosec A)
= cot² A + sec² B + 2 cot A sec B – tan² B – cosec² A + 2 tan B cosec A
= (sec² B – tan² B) – (cosec² A – cot² A) + 2(cot A sec B + tan B cosec A)
= 1 – 1 + 2(cot A sec B + tan B cosec A) … [∵ sec²B – tan² B = 1
cosec²A – cot² A = 1
= 2(cot A . sec B + tan B . cosec A) = R.H.S.

Q5: Prove the following trigonometric identities: sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A.
Ans: L.H.S. = sin A (1 + tan A) + cos A (1 + cot A)


Q6: Prove the identity:

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Q7: Prove that: (1 + cot A + tan A). (sin A – cos A) =
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Q8: Prove that:

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Q9: Find the value of:

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Q10: Prove that:  = 2 cosec A
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