If α + β + γ = π and cos α = cos β · cos γ, then the value of tan β · tan γ is _______

Correct option is D

Solution:

We are given:
α + β + γ = π
=>α = π− (β + γ)

Also given:
cos α = cos β · cos γ

Now use the identity:
cos(π − θ) = −cos θ

So:
cos α = cos(π − (β + γ)) = −cos(β + γ)

So the equation becomes:
−cos(β + γ) = cos β · cos γ
=>cos(β + γ) = −cos β · cos γ

Now use the identity:
cos(β + γ) = cos β · cos γ − sin β · sin γ

Substitute:
cos β · cos γ − sin β · sin γ = −cos β · cos γ

Bring terms together:
cos β · cos γ + cos β · cos γ = sin β · sin γ
=>2 cos β · cos γ = sin β · sin γ

Divide both sides by cos β · cos γ:

$$\frac{sin β}{cos β} \cdot \frac{sin γ}{cos γ}$$​ = tan β · tan γ

Final Answer: (D) 2