Question

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Hi thr!

My question :

Prove:
(1+cot 67°) (1+cot 65°) (1+cot 68°) (1+cot 70°)=4

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Answer 1

Answer:

Step-by-step explanation:

(1+cot 67°) (1+cot 65°) (1+cot 68°) (1+cot 70°)=4

cotA = cosA/SinA
1+CotA = 1+ CosA/SinA
1+CotA = (SinA+CosA)/SinA

(1+cot 67°) (1+cot 65°)
= (Sin 67° + Co67°)(Sin65° + Cos65°)/(Sin67°×Sin65°)
=(Sin67°Sin65° + Cos67°Cos65° + Sin67°Cos65° + Cos67°Sin65°)/(Sin67°×Sin65°)

using
SinASinB + CosACosB = Cos(A-B)
SinACosB + CosASinB = Sin(A+B)
SinASinB = (1/2)(Cos(A-B)-(Cos(A+B)
Sin67°Sin65° + Cos67°Cos65° = Cos(67-65) = Cos2
Sin67°Cos65° + Cos67°Sin65°) = Sin(67°+65°) = Sin132°

Sin67°Sin65°=(1/2)(Cos(67°-65°)-Cos(67+65))
= (1/2)(Cos2 - Cos132)

= 2(Cos2 + Sin132)/(Cos2 -Cos132)

similarly
(1+cot 68°) (1+cot 70°)
= 2(Cos2 + Sin138)/(Cos2 -Cos138)

Sin132 = - Cos138
Sin138 = - Cos132

use this and multiply and you will get 4