In a Δ ABC right angled at B, it tan A = 1/√3 , find the value of cos A cos C – sin A sin C.

Question

In a Δ ABC right angled at B, it tan A = \(\frac{1}{\sqrt3}\), find the value of cos A cos C – sin A sin C. 

(a) 1 

(b) 0 

(c) –1 

(d) 2

Answer 1

(b) 

Consider the Δ ABC in which ∠B = 90° 

Since, tan A = \(\frac{1}{\sqrt3}\)

∴ \(\frac{1}{\sqrt3}\) = \(\frac{BC}{AB}\)

⇒ \(\frac{AB}{\sqrt3}\) = \(\frac{BC}{1}\) = k

⇒ AB = √3k and BC = k

By Pythagoras’ Theorem, 

AC² = AB² + BC² 

= (√3 k)² + k² + 4k² 

⇒ AC = 2k

Now for ∠A, Base = AB, Perp. = BC, Hyp.= AC