Chapter 8: Introduction to Trigonometry & Its Equations

True

Justification:

Since, tan (90° -θ) = cot θ

False

Justification:

Since, (a²-b²) = (a+b)(a-b)

cos² 23° – sin² 67° =(cos 23°+sin 67°)(cos 23°-sin 67°)

= [cos 23°+sin(90°-23°)] [cos 23°-sin(90°-23°)]

= (cos 23°+cos 23°)(cos 23°-cos 23°) (∵sin(90°-θ) = cos θ)

= (cos 23°+cos 23°).0

= 0, which is neither positive nor negative

False

Justification:

We know that,

sin θ increases when 0° ≤ θ ≤ 90°

cos θ decreases when 0° ≤ θ ≤ 90°

And (sin 80°-cos 80°) = (increasing value-decreasing value)

= a positive value.

Therefore, (sin 80°-cos 80°) > 0.

True

Justification:

True

Justification:

According to the question,

cos A+cos² A = 1

i.e., cos A = 1- cos² A

Since,

sin² θ+cos² θ = 1

sin² θ = 1- cos² θ)

We get,

cos A = sin² A …(1)

Squaring L.H.S and R.H.S,

cos² A = sin⁴ A …(2)

To find sin²A+sin⁴ A=1

Adding equations (1) and (2),

We get

sin²A + sin⁴ A= cos A + cos² A

Therefore, sin²A+ sin⁴ A = 1

False

Justification:

L.H.S = (tan θ+2) (2 tan θ+1)

= 2 tan² θ + tan θ + 4 tan θ + 2

= 2 tan²θ+5 tan θ+2

Since, sec² θ – tan² θ = 1, we get, tan²θ = sec² θ-1

= 2(sec² θ-1) +5 tan θ+2

= 2 sec² θ-2+5 tan θ+2

= 5 tan θ+ 2 sec² θ ≠R.H.S

∴, L.H.S ≠ R.H.S