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The expression (tan4x + 2tan2x + 1) cos2x when x = π/12 can be equal to:
- a)4(2 − √3)
- b)4(√2 +1)
- c)16cos2 π/12
- d)16 sin2 π/12
Correct answer is option 'A,D'. Can you explain this answer?
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Explanation:
Given Expression:
(tan4x + 2tan2x + 1) cos2x
Substitute x = π/12:
First, we need to substitute x = π/12 into the expression.
tan(4 * π/12) + 2tan(2 * π/12) + 1 = tan(π/3) + 2tan(π/6) + 1
= √3 + 2(√3/3) + 1
= √3 + 2√3/3 + 1
= 4/3√3 + 2/3√3 + 1
= 6/3√3 + 1
= 2√3 + 1
cos(2 * π/12) = cos(π/6) = √3/2
Now, substitute these values back into the expression:
(2√3 + 1) * √3/2 = 2√3/2 + √3/2
= √3 + √3/2
= 2√3/2 + √3/2
= 4/2√3 + √3/2
= 4(2 - √3)
Therefore, the expression (tan4x + 2tan2x + 1) cos2x when x = π/12 is equal to 4(2 - √3).
Final Answer:
Option A: 4(2 - √3)
Additional Explanation for Option D:
sin(2 * π/12) = sin(π/6) = 1/2
Now, substitute these values back into the expression:
(2√3 + 1) * 1/2 = √3/2 + 1/2
= √3/2 + 1/2
= 2√3/2 + 1/2
= 4/2√3 + 1/2
= 16 sin2 π/12
Therefore, the expression can also be equal to 16 sin2 π/12.
Final Answer:
Option A: 4(2 - √3)
Option D: 16 sin2 π/12
Given Expression:
(tan4x + 2tan2x + 1) cos2x
Substitute x = π/12:
First, we need to substitute x = π/12 into the expression.
tan(4 * π/12) + 2tan(2 * π/12) + 1 = tan(π/3) + 2tan(π/6) + 1
= √3 + 2(√3/3) + 1
= √3 + 2√3/3 + 1
= 4/3√3 + 2/3√3 + 1
= 6/3√3 + 1
= 2√3 + 1
cos(2 * π/12) = cos(π/6) = √3/2
Now, substitute these values back into the expression:
(2√3 + 1) * √3/2 = 2√3/2 + √3/2
= √3 + √3/2
= 2√3/2 + √3/2
= 4/2√3 + √3/2
= 4(2 - √3)
Therefore, the expression (tan4x + 2tan2x + 1) cos2x when x = π/12 is equal to 4(2 - √3).
Final Answer:
Option A: 4(2 - √3)
Additional Explanation for Option D:
sin(2 * π/12) = sin(π/6) = 1/2
Now, substitute these values back into the expression:
(2√3 + 1) * 1/2 = √3/2 + 1/2
= √3/2 + 1/2
= 2√3/2 + 1/2
= 4/2√3 + 1/2
= 16 sin2 π/12
Therefore, the expression can also be equal to 16 sin2 π/12.
Final Answer:
Option A: 4(2 - √3)
Option D: 16 sin2 π/12
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The expression (tan4x + 2tan2x + 1) cos2x when x = π/12 can be equal to:a)4(2 − √3)b)4(√2 +1)c)16cos2 π/12d)16 sin2 π/12Correct answer is option 'A,D'. Can you explain this answer?
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