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Number of real solutions of the equation (tan x + 1)(tan x + 3)(tan x + 5)(tan x + 7) = 33
  • a)
    will be 2 in the interval [-π/2, π/2]
  • b)
    will be 4 in the interval [-π/2, π/2]
  • c)
    will be 3 in the interval [-π/2, π]
  • d)
    will be 4 in the interval [-π/2,π]
Correct answer is option 'A,D'. Can you explain this answer?
Most Upvoted Answer
Number of real solutions of the equation (tan x + 1)(tan x + 3)(tan x...
(tan x + 1)(tan x + 3)(tan x + 5)(tan x + 7) = 33
(tan2 x + 8 tan x + 15)(tan2 x + 8 tan x + 7) = 33
Let tan2 x + 8 tan x = p.
∴ (p + 15)(p + 7) = 33
⇒ p2 + 22p + 72 = 0
⇒ p = -18 (rejected) or p = -4
⇒ tan2 x + 8 tan x + 4 = 0
So,
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Community Answer
Number of real solutions of the equation (tan x + 1)(tan x + 3)(tan x...
Explanation:

Given Equation:
(tan x + 1)(tan x + 3)(tan x + 5)(tan x + 7) = 33

Step 1: Analyzing the Equation
- The given equation represents a product of four different tangent functions.
- We need to find the number of real solutions for this equation within the given interval.

Step 2: Finding the Roots
- In the interval [-π/2, π/2], the function tan x is defined for values of x where -π/2 < x="" />< />
- By analyzing the equation, we can see that the product of four tangent functions will result in a constant (33).
- This implies that there will be two pairs of solutions where the tangent functions cancel each other out resulting in real solutions.

Step 3: Conclusion
- Therefore, the equation will have 2 real solutions in the interval [-π/2, π/2].

Step 4: Additional Analysis
- If we consider the interval [-π/2, π], there will be 4 real solutions for the given equation.
- This is because the tangent function has a period of π and repeats every π units. Hence, we will have 4 real solutions within this extended interval.

Final Answer:
- The number of real solutions of the equation will be 2 in the interval [-π/2, π/2] and 4 in the interval [-π/2, π]. Hence, the correct answers are options A and D.
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Number of real solutions of the equation (tan x + 1)(tan x + 3)(tan x + 5)(tan x + 7) = 33a)will be 2 in the interval [-π/2, π/2]b)will be 4 in the interval [-π/2, π/2]c)will be 3 in the interval [-π/2, π]d)will be 4 in the interval [-π/2,π]Correct answer is option 'A,D'. Can you explain this answer?
Question Description
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