Question

The value of $\tan 7x\tan 3x\tan 4x\, = $
$\left( A \right)\,\tan 7x - \tan 3x - \tan 4x$
$\left( B \right)\,\dfrac{{\sin 7x - \tan 3x - \sin 4x}}{{\cos 7x - \cos 3x - \cos 4x}}$
$\left( C \right)\,0$
$\left( D \right)\,None\,of\,these$

Answer 1

Hint: Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. Formula used: $\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$

Complete step-by-step answer:
In the given question,
We know that
$7x = 3x + 4x$
Taking tan both sides
$\tan \left( {7x} \right) = \tan \left( {3x + 4x} \right)$
Using formula,$\tan \left( {A + B} \right) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$
$\tan \left( {7x} \right) = \dfrac{{\tan \left( {3x} \right) + \tan \left( {4x} \right)}}{{1 - \tan \left( {3x} \right)\tan \left( {4x} \right)}}$
On cross multiplication, we get
$\tan 7x\left( {1 - \tan 3x\tan 4x} \right) = \tan 3x + \tan 4x$
Now, multiplying in L.H.S
$\tan 7x - \tan 7x\tan 3x\tan 4x = \tan 3x + \tan 4x$
On transposing, we get
$\tan 7x - \tan 3x - \tan 4x = \tan 7x\tan 3x\tan 4x$
Therefore, the required answer is $\tan 7x - \tan 3x - \tan 4x.$
So, the correct answer is “Option a”.

Note: The domain of the function y=tan(x) ) is all real numbers except the values where cos(x) is equal to 0 , that is, the values π2+πn for all integers n . The range of the tangent function is all real numbers.