Question

Prove that $cos^25+cos^210+cos^215+...+cos^290=17/2$

Answer 1

Answer:

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Answer 2

Question :- Prove that , cos²5 + cos²10 + cos²15 + ______________ cos²90 = (17/2)

Solution :-

Solving LHS,

→ cos²5 + cos²10 + cos²15 + ______________ cos²90

Making pairs ,

→ (cos²5 + cos²85) + (cos²10 + cos²80) + (cos²15 + cos²75) + (cos²20 + cos²70) + (cos²25 + cos²65) + (cos²30 + cos²60) + (cos²35 + cos²55) + (cos²40 + cos²50) + cos²45 + cos²90

Now, we know that, cosA = sin(90 - A)

So,

1. cos 85 = sin(90 - 85) = sin 5
2. cos 80 = sin(90 - 80) = sin 10
3. cos 75 = sin(90 - 75) = sin 15

• Similarly,
• cos 50 = sin(90 - 50) = sin 40

Putting these values we get,

→ (cos²5 + sin²5) + (cos²10 + sin²10) + (cos²15 + sin²15) + (cos²20 + sin²20) + (cos²25 + sin²25) + (cos²30 + sin²30) + (cos²35 + sin²35) + (cos²40 + sin²40) + cos²45 + cos²90

Now, we know that, cos²A + sin²A = 1 ,

Therefore,

→ 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + cos²45 + cos²90

Now, Putting ,

• cos 45 = (1/√2)
• cos 90 = 0

→ 8 + (1/√2)² + 0

→ 8 + (1/2)

→ (16 + 1)/2

→ (17/2) = RHS. (Hence, Proved).

Learn More :-

prove that cosA-sinA+1/cos A+sinA-1=cosecA+cotA

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