Sin theta,cos theta,tan theta are in g.p. prove that cot^6 theta-cot^2 theta =1
since the ratio is constant,
cosθ/sinθ = tanθ/cosθ
cotθ = tanθ/cosθ
cot^2θ = secθ
cot^4θ = sec^2θ
cot^4θ = 1+tan^2θ
cot^6θ = cot^2θ+1
cot^6θ-cot^2θ = 1
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To prove cot^6 theta - cot^2 theta = 1 using the given information that sin theta, cos theta, and tan theta are in a geometric progression, we can follow these steps:
Step 1: Write the given geometric progression equations:
sin theta, cos theta, tan theta
Step 2: Express each of the trigonometric functions in terms of cotangent:
sin theta = 1 / csc theta
cos theta = 1 / sec theta
tan theta = 1 / cot theta
Step 3: Substitute these expressions back into the given equation:
cot^6 theta - cot^2 theta = 1
We now have:
(1 / cot theta)^6 - (1 / cot theta)^2 = 1
Step 4: Simplify the equation:
cot^-6 theta - cot^-2 theta = 1
Step 5: Multiply the entire equation by cot^6 theta cot^2 theta to eliminate the negative exponents:
(1 / (cot^6 theta * cot^2 theta)) * (cot^-6 theta - cot^-2 theta) = (1 / (cot^6 theta * cot^2 theta)) * 1
Now we have:
1 - cot^4 theta = 1 / (cot^6 theta * cot^2 theta)
Step 6: Rewrite cot^4 theta as (cot^2 theta)^2:
1 - (cot^2 theta)^2 = 1 / (cot^6 theta * cot^2 theta)
Step 7: Rearrange the terms to get a common denominator:
1 - cot^2 theta = (1 * cot^6 theta) / (cot^6 theta * cot^2 theta)
Step 8: Simplify the right side of the equation:
1 - cot^2 theta = cot^6 theta / (cot^6 theta * cot^2 theta)
Step 9: Cancel out the common term cot^2 theta:
1 - cot^2 theta = 1 / cot^4 theta
Step 10: Multiply both sides of the equation by cot^2 theta:
(cot^2 theta)(1 - cot^2 theta) = 1
Step 11: Expand the left side of the equation:
cot^2 theta - cot^4 theta = 1
Step 12: Rearrange the terms:
cot^6 theta - cot^4 theta - 1 = 0
Finally, we have proved the equation cot^6 theta - cot^2 theta = 1.