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:
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:
\(-\frac{11\pi}{6}\)
\(\frac{\pi}{10}\)
\(-\frac{7\pi}{30}\)
\(-\frac{\pi}{15}\)
The Correct Option is A
Solution and Explanation
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
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