Question

The value of \[ \sin(x+y)\sec x\sec y \] is

• A. \(\cos x\cos y\)
• B. \(\tan x-\tan y\)
• C. \(\cos x+\cos y\)
• D. \(\tan x+\tan y\)

Hint:

To simplify trigonometric products involving \(\sec x\) and \(\sec y\), first expand compound angles and then convert \(\sec\) into reciprocal cosine.

Answer 1

The Correct Option is D

Step 1: Expand sin(x+y) and multiply by sec x sec y.
\(\sin(x+y)\sec x\sec y=(\sin x\cos y+\cos x\sin y)\cdot\frac{1}{\cos x\cos y}\).

Step 2: Simplify each term.
\(=\frac{\sin x}{\cos x}+\frac{\sin y}{\cos y}=\tan x+\tan y\). \[ \boxed{\tan x+\tan y} \]