Question

Tan (45+a) +tan(45-a) = 3 find tan^2(45+a) + tan^2(45-a)

Options :A)4
B)5
C)6
D)7

Answer 1

Question :- (Nice Question).

• if tan(45+a) +tan(45-a) = 3 find tan²(45+a) + tan²(45-a) = ?

Formula used :-

• tan(A + B) = (tanA + tanB) / (1 - tanA* tanB)
• tan(A - B) = (tanA - tanB) / (1 + tanA*tanB)
• Tan45° = 1
• (a + b)² = a² + b² + 2ab .

Solution :-

→ Tan(45+a) + tan(45-a) = 3

Squaring Both sides we get,

→ [Tan(45+a) + tan(45-a)]² = 3²

Using (a + b)² = a² + 2ab + b² in LHS,

→ tan²(45+a)+ 2*tan(45+a)*tan(45-a) + tan²(45-a) = 9

__________________________

Lets Try to Solve 2*tan(45+a)*tan(45-a) First :-

→ 2*tan(45+a)*tan(45-a)

using tan(A + B) & tan(A - B) formula told above we get,

→ 2 * [ (tan45 + tana) / (1 - tan45* tana) ] * [(tan45 - tana) / (1 + tan45* tana)]

Putting tan45° = 1 now,

→ 2 * [ (1 + tana) / (1 - tana) ] * [(1 - tana) / (1 + tana)]

→ 2 * 1

→ 2

___________________

Putting This value now, we get :-

→ tan²(45+a) + 2 + tan²(45 - a) = 9

→ tan²(45+a) + tan²(45 - a) = 9 - 2

→ tan²(45+a) + tan²(45 - a) = 7 . (Ans). (Option D).

Answer 2

|| ✰✰ ANSWER ✰✰ ||

Now,

Square both sides of the starting equation:-

tan ( 45 + a ) + tan ( 45 - a ) = 3

Now, we have:-

tan2(45+x)+ 2 tan(45+x)tan(45-x) + tan2(45-x)=32

Two trig formulas which will be handy are:-

tan(u + v) = (tan u + tan v) / (1 - tan u tan v)

tan(u - v) = (tan u - tan v) / (1 + tan u tan v)

Now, look at the middle term:-

2 tan(45+x)tan(45-x)

Also recall tan 45 = 1

By simplifying using the two trig formulas above , Find that :-

2 tan(45+x)tan(45-x) = 2

Then going back to the starting point we have:-

tan2(45+x)+ 2 tan(45+x)tan(45-x) + tan2(45-x)=32

tan2(45+x)+ 2 + tan2(45-x)=9

Subtracting 2 from both sides we get :-

tan2(45+x)+ tan2(45-x)=7

✪✪ Hence Proved ✪✪

hope it helps you mark as brainliest please