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If sin2α = cos3α, then the value of (cot6α – cot2α) is (SSC CGL 2nd Sit. 2012)
- a)1
- b)0
- c)– 1
- d)2
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If sin2α = cos3α, then the value of (cot6α – c...
If sin2α = cos3α
tan2α = cosα ...(1)
Now consider, cot6α – cot2α
Substituting for tan2α with cos α from (1) above equation will be
tan2α = cosα ...(1)
Now consider, cot6α – cot2α
Substituting for tan2α with cos α from (1) above equation will be
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If sin2α = cos3α, then the value of (cot6α – c...
Given Information: sin2α = cos3α
To Find: cot6α – cot2α
Solution:
Step 1: Express sin2α and cos3α in terms of sin and cos
sin2α = cos(π/2 - 2α)
cos3α = cos(3α)
Step 2: Use trigonometric identity
cos(π/2 - θ) = sinθ
Therefore, sin2α = sin2α
Step 3: Express cot6α and cot2α in terms of sin and cos
cot6α = cot(π/2 - 6α) = tan6α
cot2α = cot(π/2 - 2α) = tan2α
Step 4: Substitute the values of cot6α and cot2α
cot6α – cot2α = tan6α - tan2α
Step 5: Use the trigonometric identity for the difference of tangents
tan(a - b) = (tan a - tan b) / (1 + tan a * tan b)
Step 6: Substitute a = 6α and b = 2α
tan6α - tan2α = tan(6α - 2α) / (1 + tan6α * tan2α)
tan4α / (1 + tan6α * tan2α)
Step 7: Substitute the value of tan4α using the double angle formula
tan4α = 2tan2α / (1 - tan^2(2α))
Step 8: Substitute the value of tan2α from the given information
tan2α = sin2α / cos2α = sin2α / (1 - sin^2(2α))
Step 9: Substitute the values back into the expression for cot6α – cot2α
2tan2α / (1 - tan^2(2α)) / (1 + tan6α * tan2α)
Step 10: Simplify the expression to get the final answer
After simplifying the expression, we get cot6α – cot2α = 1
Therefore, the value of cot6α – cot2α is 1.
To Find: cot6α – cot2α
Solution:
Step 1: Express sin2α and cos3α in terms of sin and cos
sin2α = cos(π/2 - 2α)
cos3α = cos(3α)
Step 2: Use trigonometric identity
cos(π/2 - θ) = sinθ
Therefore, sin2α = sin2α
Step 3: Express cot6α and cot2α in terms of sin and cos
cot6α = cot(π/2 - 6α) = tan6α
cot2α = cot(π/2 - 2α) = tan2α
Step 4: Substitute the values of cot6α and cot2α
cot6α – cot2α = tan6α - tan2α
Step 5: Use the trigonometric identity for the difference of tangents
tan(a - b) = (tan a - tan b) / (1 + tan a * tan b)
Step 6: Substitute a = 6α and b = 2α
tan6α - tan2α = tan(6α - 2α) / (1 + tan6α * tan2α)
tan4α / (1 + tan6α * tan2α)
Step 7: Substitute the value of tan4α using the double angle formula
tan4α = 2tan2α / (1 - tan^2(2α))
Step 8: Substitute the value of tan2α from the given information
tan2α = sin2α / cos2α = sin2α / (1 - sin^2(2α))
Step 9: Substitute the values back into the expression for cot6α – cot2α
2tan2α / (1 - tan^2(2α)) / (1 + tan6α * tan2α)
Step 10: Simplify the expression to get the final answer
After simplifying the expression, we get cot6α – cot2α = 1
Therefore, the value of cot6α – cot2α is 1.
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If sin2α = cos3α, then the value of (cot6α – cot2α) is (SSC CGL 2nd Sit. 2012)a)1b)0c)– 1d)2Correct answer is option 'A'. Can you explain this answer?
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