Question

Prove that: tan65 = tan25 + 2tan40

Answer 1

tan65 - tan40 = tan 25 + tan 40 . using tanA - tan B =sin(A - B)/cosAcosB and tanA + tan B = sin( A + B)/cosAcosB we get :: to prove sin 25/ cos65 = sin65 / cos25 sin 25 = cos (90 - 25) = cos 65 and sin 65 = cos (90 - 65) = cos 25

Answer 2

To prove:

tan65 = tan25 + 2tan40

Solution:

The given first we have to take LHS or RHS then apply the formula and simplify the terms to get another side.

Now taking LHS  tan 65 To expand this form apply the formula of the trigonometric identities  

$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$

$\Rightarrow \tan A-\tan B=\tan (A-B)(1+\tan A \cdot \tan B)$

Let A = 65, B = 25

$\tan 65-\tan 25=\tan (65-25)(1+\tan 65 . \tan 25)$

$\tan 65-\tan 2=\tan (40)((1+\tan 65) \tan (90-65))$

$\tan 65-\tan 25=\tan (40)(1+\tan 65 . \cot 65)$

$tan65- tan25 = 2tan(40)$

$\bold{tan65 = tan25+2 tan40}$

Hence, it is proved.