Trigonometrical Identities — Assertion-Reason Type Questions

Assertion (A): If sin² A + sin A = 1 then cos⁴ A + cos² A = 1.

Reason (R): 1 - sin² A = cos² A

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

⇒ sin² A + sin A = 1

⇒ sin A = 1 - sin² A

⇒ sin A = cos² A

Squaring both the sides, we get :

⇒ (sin A)² = (cos² A)²

⇒ sin² A = cos⁴ A ......................(1)

Solving L.H.S. of cos⁴ A + cos² A = 1, we get :

⇒ cos⁴ A + cos² A

⇒ sin² A + cos² A [From equation (1)]

⇒ 1.

Since, L.H.S. = R.H.S.

∴ Assertion (A) is true.

As we know that, sin² A + cos² A = 1

⇒ 1 - sin² A = cos² A

∴ Reason (R) is true.

∴ Both (A) and (R) are true, and (R) is the correct explanation for (A).

Hence, option 1 is the correct option.

Assertion (A): $\Big(\dfrac{1 + \tan \theta}{1 + \cot \theta} \Big)^2 = \tan^2 \theta$

Reason (R): tan² θ + sec² θ = 1

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

$\Rightarrow \Big(\dfrac{1 + \tan \theta}{1 + \cot \theta} \Big)^2 \\[1em] \Rightarrow \Big(\dfrac{1 + \tan \theta}{1 + \dfrac{1}{\tan \theta}} \Big)^2 \\[1em] \Rightarrow \Big(\dfrac{1 + \tan \theta}{ \dfrac{\tan \theta + 1}{\tan \theta}} \Big)^2 \\[1em] \Rightarrow \tan^2 \theta$

∴ Assertion (A) is true.

tan² θ + sec² θ = 1 is incorrect; the correct identity is sec² θ − tan² θ = 1.

∴ Reason (R) is false.

Hence, option 3 is the correct option.

Assertion (A): (1 − cosec² θ)(1 − sec² θ) = 1

Reason (R): 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Solving L.H.S,

(1 − cosec² θ)(1 − sec² θ)

= (− cot² θ)(− tan² θ) [∵ 1 − cosec² θ = − cot² θ and 1 − sec² θ = − tan² θ]

= $\dfrac{1}{\tan^2 \theta}$ (tan² θ) = 1

∴ Assertion (A) is true.

1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ are standard identities, and these are exactly the relations used to simplify the Assertion.

∴ Reason (R) is true and is the correct explanation of A.

Hence, option 1 is the correct option.

Assertion (A): If sec θ + tan θ = p, then sec θ = $\dfrac{2p}{p^2 + 1}$

Reason (R): sec² θ − tan² θ = 1

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given, sec θ + tan θ = p.

We know that sec² θ − tan² θ = 1.

⇒ (sec θ + tan θ)(sec θ − tan θ) = 1

⇒ (p)(sec θ − tan θ) = 1

⇒ sec θ − tan θ = $\dfrac{1}{p}$

Adding sec θ + tan θ = p and sec θ − tan θ = $\dfrac{1}{p}$ :

⇒ 2 sec θ = p + $\dfrac{1}{p}$ = $\dfrac{p^2 + 1}{p}$

⇒ sec θ = $\dfrac{p^2 + 1}{2p}$.

So the correct value is $\dfrac{p^2 + 1}{2p}$, not $\dfrac{2p}{p^2 + 1}$.

∴ Assertion (A) is false.

sec² θ − tan² θ = 1 is a standard Pythagorean identity.

∴ Reason (R) is true.

Hence, option 4 is the correct option.

Assertion (A): For an acute angle θ, if sin θ = cos θ, then 2 sin² θ + tan² θ = 2

Reason (R): For any acute angle θ, sin (90° − θ) = cos θ

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given, sin θ = cos θ.

⇒ $\dfrac{\sin \theta}{\cos \theta}$ = 1 ⇒ tan θ = 1, which for an acute angle gives θ = 45°.

Also, sin θ = cos θ ⇒ sin² θ = cos² θ, and since sin² θ + cos² θ = 1, we get 2 sin² θ = 1.

⇒ 2 sin² θ + tan² θ = 1 + (1)² = 1 + 1 = 2.

∴ Assertion (A) is true.

sin (90° − θ) = cos θ is a true complementary-angle identity.

∴ Reason (R) is true.

However, the value 2 sin² θ + tan² θ = 2 is obtained using tan θ = 1 and 2 sin² θ = 1, not from the complementary-angle relation. So R does not explain A.

Hence, option 2 is the correct option.

Assertion (A): If sec θ + tan θ = a and sec θ − tan θ = b then ab = 1.

Reason (R): sec² θ - tan² θ = 1

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

⇒ sec θ + tan θ = a

⇒ sec θ − tan θ = b

⇒ ab = (sec θ + tan θ)(sec θ - tan θ)

⇒ ab = sec² θ - tan² θ

⇒ ab = 1

So assertion (A) is true.

We know that,

⇒ sec² θ - tan² θ = 1

This is a fundamental trigonometric identity.

So reason (R) is true.

Thus, Both (A) and (R) are true and (R) is the correct explanation of (A).

Hence, option 1 is the correct option.