Question

Anyone plz answer this question ??​

Answer 1

Solution :-

First Lets Prove a Identity :-

→ {1/(1 + tan³θ)} + {1/(1 + cot³θ)}

Putting cotθ = (1/tanθ) we get,

→ {1/(1 + tan³θ)} + [ 1 + /{ 1 + (1/tan³θ)} ]

Taking LCM of Second Part ,

→ {1/(1 + tan³θ)} + [ 1 + / { (tan³θ + 1) / tan³θ } ]

→ {1/(1 + tan³θ)} + {tan³θ / (1 + tan³θ)}

Taking LCM on Both Now,

→ (1 + tan³θ) / (1 + tan³θ)

→ 1.

So, we can Conclude That,

☛ {1/(1 + tan³θ)} + {1/(1 + cot³θ)} = 1

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Question :-

➺ 1/(1+tan³10°) + 1/(1+tan³20°) + 1/(1+tan30°) ______________ 1/(1+tan³80°)

we know That,

☞ tan(90-θ) = cotθ

So,

☞ tan80° = tan(90°-10°) = cot10°

☞ tan70° = tan(90°-20°) = cot20°

☞ tan60° = tan(90°-30°) = cot30°

☞ tan50° = tan(90-40°) = cot40°

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Putting These values , we can say That, we have To Find Value of :-

➼ 1/(1+tan³10°) + 1/(1+tan³20°) + 1/(1+tan³30°) + 1/(1+tan³40°) + 1/(1+cot³40°) + 1/(1+cot³30°) + 1/(1+cot³20°) + 1/(1+cot³10°)

Re - arranging Them now, we get,

➼ [ 1/(1+tan³10°) + 1/(1+cot³10°) ] + [ 1/(1+tan³20°) + 1/(1+cot³20°) ] + [ 1/(1+tan³30°) + 1/(1+cot³30°) ] + [ 1/(1+tan³40°) + 1/(1+cot³40°) ]

Using Our Proving Identity Now , we get,

☛ 1 + 1 + 1 + 1

☛ 4 (Ans).

Answer 2

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$\huge\tt{SOLUTION:}$

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Proving the identities-

↪{1/(1+tan³∅)}+{1/(1+cot³∅)}

↪{1/(1+tan³∅)}+[1/{(tan³∅)}]

↪{1/(1+tan³∅)}+[1+/{(tan³∅)}]

↪{1/(1+tan³∅)}+[1+/{tan³∅+1)/(tan³∅)}]

↪{1/(1+tan³∅)}+[tan³∅/(1+tan³∅)]

↪(1+tan³∅)/(1+tan³∅)

↪1

We can conclude that,

↪{1/(1+tan³∅)}+{1/(1+cot³∅)}=1

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$\huge\tt{QUESTION:}$

1/(1+tan³10°)+(1+tan³20°)+1/(1+tan30°)_____1/(1+tan³80°)

As we know,tan(90-∅)=cot∅

Then,

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➡tan80° = {tan90°-10°} = cot10°

➡tan70° = {tan90°-20°} = cot20°

➡tan60° = {tan90°-30°} = cot30°

➡tan50° = {tan90°-40°} = cot40°

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$\huge\tt{SOLVING:}$

By putting the given values here-

➡1/(tan³10°)+1/(1+tan³20°)+1/(1+tan³30°)+1/(1+tan³40°)+1/(1cot³30°)+1/(1+cot³20°)+1/(1+cot³10°)

➡ [1/(1+tan³10°)+1/(1+cot³10°)+1/(1+tan³20°)+1/(1+cot³20°)+1/(1+tan³30°)+1/(1cot³30°)+(1+tan³40°)+(1+cot³40°)]

Now,by using our above identity,,we get-

➡ 1+1+1+1

⏭ 4

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