Assertion (A): For an acute angle θ, if sin θ = cos θ, then 2 sin² θ + tan² θ = 1

Reason (R): For any acute angle θ, sin (90° − θ) = cos (45° + θ)

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

For acute angle,

$\dfrac{\sin \theta}{\cos \theta}$ = 1

tan θ = 1

For an acute angle, this occurs when $\theta = 45°$.

Substitute theta in expression,

⇒ 2 sin² θ + tan² θ = 1

⇒ 2 sin² 45° + tan² 45° = 1

$\Rightarrow 2\Big(\dfrac{1}{\sqrt2}\Big)^2 + 1 \\[1em] \Rightarrow 2\Big(\dfrac{1}{2}\Big) + 1 \\[1em] \Rightarrow 1 + 1 \\[1em] \Rightarrow 2.$

Assertion is false.

sin (90° − θ) = cos (45° + θ) is not true.

The standard trigonometric identity is: sin (90° − θ) = cos θ

Reason is false

Both A and R are false

Hence, option 4 is the correct option.