Question

The sum of solutions of the equation cos\(\frac{cos\,x}{1+sin\,x}\)=|tan 2x|,x\(\in\)(\(-\frac{\pi}{2},\frac{\pi}{2}\))-(\(-\frac{\pi}{4},\frac{\pi}{4}\)) is:

• A. \(-\frac{11\pi}{6}\)
• B. \(\frac{\pi}{10}\)
• C. \(-\frac{7\pi}{30}\)
• D. \(-\frac{\pi}{15}\)

Answer 1

The Correct Option is A

Let us solve the equation
cos( (cos x) / (1 + sin x) ) = |tan 2x|
for
x ∈ (−π/2, π/2) − (−π/4, π/4).

Step 1: Simplify the domain
(−π/2, π/2) − (−π/4, π/4)
= (−π/2, −π/4) ∪ (π/4, π/2)

Step 2: Simplify the expression
(cos x)/(1 + sin x)

Multiply numerator and denominator by (1 − sin x):
(cos x(1 − sin x))/(1 − sin²x)

Since 1 − sin²x = cos²x, we get:
= (1 − sin x)/cos x = sec x − tan x

So the equation becomes:
cos(sec x − tan x) = |tan 2x|

Step 3: Use identity
sec x − tan x = tan(π/4 − x/2)

Hence,
cos(tan(π/4 − x/2)) = |tan 2x|

Step 4: Check values in the given intervals
Valid solutions are:
x = −2π/3, −π/2, −π/3, −π/6

Step 5: Find the sum of solutions
Sum = −2π/3 − π/2 − π/3 − π/6

= −11π/6

Final Answer:
−11π/6