Q1:
Ans: L.H.S. =
Q2: Prove that
Ans: L.H.S. =
Q3: Prove that
Ans: L.H.S. =
Q4: Prove that
Ans: L.H.S =
Q5: Find the value of:
(i) sin 75°
(ii) tan 15°
Ans: (i) sin 75° = sin (45°+ 30°)
= sin 45° cos 30° + cos 45° sin 30°
[sin (x + y) = sin x cos y + cos x sin y]
(ii) tan 15° = tan (45° – 30°)
Q6: Prove that:
Ans:
Q7: Prove that:
Ans: It is known that
∴ L.H.S. =
Q8: Prove that
Ans:
Q9:
Ans: L.H.S. =
Q10: Prove that sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
Ans: L.H.S. = sin (n + 1)x sin(n + 2)x + cos (n + 1)x cos(n + 2)x
Q11: Prove that
Ans: It is known that .
∴L.H.S. =
Q12: Prove that sin^{2} 6x – sin^{2} 4x = sin 2x sin 10x
Ans: It is known that
∴ L.H.S. = sin^{2}6x – sin^{2}4x
= (sin 6x + sin 4x) (sin 6x – sin 4x)
= (2 sin 5x cos x) (2 cos 5x sin x)
= (2 sin 5x cos 5x) (2 sin x cos x)
= sin 10x sin 2x
= R.H.S.
Q13: Prove that cos^{2} 2x – cos^{2} 6x = sin 4x sin 8x
Ans: It is known that
∴ L.H.S. = cos^{2} 2x – cos^{2} 6x
= (cos 2x + cos 6x) (cos 2x – 6x)
= [2 cos 4x cos 2x] [–2 sin 4x (–sin 2x)]
= (2 sin 4x cos 4x) (2 sin 2x cos 2x)
= sin 8x sin 4x
= R.H.S.
Q14: Prove that sin 2x + 2sin 4x + sin 6x = 4cos^{2} x sin 4x
Ans: L.H.S. = sin 2x + 2 sin 4x + sin 6x
= [sin 2x + sin 6x] +2 sin 4x
= 2 sin 4x cos (– 2x)+ 2 sin 4x
= 2 sin 4x cos 2x + 2 sin 4x
= 2 sin 4x (cos 2x + 1)
= 2 sin 4x (2 cos^{2} x – 1+ 1)
= 2 sin 4x (2 cos^{2} x)
= 4cos^{2} x sin 4x
= R.H.S.
Q15: Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Ans: L.H.S = cot 4x (sin 5x sin 3x)
= 2 cos 4x cos x
R.H.S. = cot x (sin 5x – sin 3x)
= 2 cos 4x. cos x
L.H.S. = R.H.S.
Q16: Prove that
Ans: It is known that
∴ L.H.S =
Q17: Prove that
Ans: It is known that
∴L.H.S. =
Q18: Prove that
Ans: It is known that
∴ L.H.S. =
Q19: Prove that
Ans: It is known that
∴L.H.S. =
Q20: Prove that
Ans: It is known that
∴L.H.S. =
Q21: Prove that
Ans: L.H.S. =
Q22: Prove that cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Ans: L.H.S. = cot x cot 2x – cot 2x cot 3x – cot 3x cot x
= cot x cot 2x – cot 3x (cot 2x + cot x)
= cot x cot 2x – cot (2x + x) (cot 2x + cot x)
= cot x cot 2x – (cot 2x cot x – 1)
= 1 = R.H.S.
Q23: Prove that
Ans: It is known that .
∴ L.H.S. = tan 4x = tan 2(2x)
Q24: Prove that cos 4x = 1 – 8sin^{2 }x cos^{2 }x
Ans: L.H.S. = cos 4x
= cos 2(2x)
= 1 – 2 sin^{2} 2x [cos 2A = 1 – 2 sin^{2} A]
= 1 – 2(2 sin x cos x)^{2} [sin2A = 2sin A cosA]
= 1 – 8 sin^{2}x cos^{2}x
= R.H.S.
Q25: Prove that: cos 6x = 32 cos^{6} x – 48 cos^{4} x + 18 cos^{2} x – 1
Ans: L.H.S. = cos 6x
= cos 3(2x)
= 4 cos^{3} 2x – 3 cos 2x [cos 3A = 4 cos^{3} A – 3 cos A]
= 4 [(2 cos^{2} x – 1)^{3} – 3 (2 cos^{2} x – 1) [cos 2x = 2 cos^{2} x – 1]
= 4 [(2 cos^{2} x)^{3} – (1)^{3} – 3 (2 cos^{2} x)^{2} + 3 (2 cos^{2} x)] – 6cos^{2} x + 3
= 4 [8cos^{6}x – 1 – 12 cos^{4}x + 6 cos^{2}x] – 6 cos^{2}x + 3
= 32 cos^{6}x – 4 – 48 cos^{4}x + 24 cos^{2} x – 6 cos^{2}x + 3
= 32 cos^{6}x – 48 cos^{4}x + 18 cos^{2}x – 1
= R.H.S.
209 videos443 docs143 tests

1. What are the basic trigonometric functions? 
2. How are trigonometric functions used in reallife applications? 
3. What is the unit circle and how is it related to trigonometric functions? 
4. How do you find the values of trigonometric functions for angles beyond 90 degrees? 
5. Can trigonometric functions be used to solve for unknown sides and angles in a triangle? 
209 videos443 docs143 tests


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