Which of the following is true for all values.

Which of the following is true for all values of θ (0° < θ < 90°):
(a) cos² θ – sin² θ = 1
(b) cosec² θ – sec² θ = 1
(c) sec² θ – tan² θ = 1
(d) cot² θ – tan² θ = 1

Solution:

∴ sec² θ – tan² θ = 1 is true for all values of θ as it is an identity.
(0° < θ < 90°) (c)

If θ is an acute angle of a right triangle, then the value of sin θ cos (90° – θ) + cos θ sin (90° – θ) is
(a) 0
(b) 2 sin θ cos θ
(c) 1
(d) 2 sin² θ

Solution:

sin θ cos (90° – θ) + cos θ sin (90° – θ)
= sin θ sin θ + cos θ cos θ
{ ∵ sin(90° – θ) = cosθ, cos (90° – θ) = sin θ }
= sin² θ + cos² θ = 1 (c)

The value of cos 65° sin 25° + sin 65° cos 25° is
(a) 0
(b) 1
(b) 2
(d) 4

Solution:

cos 65° sin 25° + sin 65° cos 25°
= cos (90° – 25°) sin 25° + sin (90° – 25°) cos 25°
= sin 25° . sin 25° + cos 25° . cos 25°
= sin² 25° + cos² 25°
( ∵ sin² θ + cos² θ = 1)
= 1 (b)

More Solutions:

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• Solve: sec4 A (1 – sin4 A) – 2 tan2 A = 1
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• Solve: 2 (sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ) + 1 = θ