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How to prove this tangent identities?

Hi, brilliant users. I have a little problem again about trigonometry. it's all about tangent function. Here's the problem:

Prove that: tan 5.tan 40 - tan 10.tan 35 = tan 10 + tan 35 - tan 5 - tan 40

From where, I must started first? thanks

Note by Leonardo Chandra
3 years, 9 months ago

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Hint: with, \(A+B = 45^\circ \), we have \( 1 = \tan (A+B) = \frac { \tan (A) + \tan (B) } {1 - \tan (A) \tan (B) } \) Pi Han Goh · 3 years, 9 months ago

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@Pi Han Goh Thanks. I got it now. The key is your hint and operation at the right side:

We got: tan 10+tan 35 = tan 45 (1-tan 10.tan 35) and tan 5+tan 40= tan 45(1-tan 5.tan 40)

So, the equation become:

tan 10+tan 35-(tan 5+tan 40)

= 1-tan 10.tan 35-(1-tan 5.tan 40)

= -tan 10.tan 35 +tan 5.tan 40

= tan 5.tan 40 - tan 10.tan 35 (proved)

Thanks, Pi Han :) Leonardo Chandra · 3 years, 9 months ago

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Tan40°+tan5°+tan40°×tan5°=1 prove it Rahul Prajapati · 1 year, 11 months ago

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