Question

Solve $\tan 5x\tan 3x\tan 2x = $
A. $\tan 5x - \tan 3x - \tan 2x$
B. $(\sin 5x - \sin 3x - \sin 2x)/(cos5x - \cos 3x - \cos 2x)$
C. 0
D. None of these

Answer 1

Hint: In order to answer this question, first we will write any of the algebraic equations in which we can include the trigonometric term $\tan $ i.e.. $2x = 5x - 3x$. Then after taking $\tan $ on both sides, we will solve it to get the final solution.

Formula used:
We will also use the formula during solving:
$\tan (a - b) = \dfrac{{\tan a - \tan b}}{{1 + \tan a.\tan b}}$

Complete step by step answer:
The given trigonometric expression is: $\tan 5x\tan 3x\tan 2x$
Now, we can write:
$2x = 5x - 3x$
(when we subtract $3x$ from $5x$ , then we get $2x$, as we know this simply)
Now, taking $\tan $ in both sides of the above equation:
$ \Rightarrow \tan 2x = \tan (5x - 3x)$
[as we know the formula: $\tan (a - b) = \dfrac{{\tan a - \tan b}}{{1 + \tan a.\tan b}}$ ]
$ \Rightarrow \tan 2x = \dfrac{{\tan 5x - \tan 3x}}{{1 + \tan 5x\tan 3x}}$
Now, we will cross multiply because in R.H.S, we can’t divide further:
$\tan 2x(1 + \tan 5x\tan 3x) = \tan 5x - \tan 3x \\
\Rightarrow \tan 2x + \tan 5x\tan 3x\tan 2x = \tan 5x - \tan 3x \\
\therefore \tan 5x\tan 3x\tan 2x = \tan 5x - \tan 3x - \tan 2x \\ $
Therefore, $\tan 5x\tan 3x\tan 2x$ is equal to $\tan 5x - \tan 3x - \tan 2x$ .

Hence, the correct option is A.

Note: The steps to solve the trigonometric identities or expression are as follows-
Step-1: Convert all sec, cosec, cot, and tan to sin and cos. Most of this can be done using the quotient and reciprocal identities.
Step-2: Check all angles for sums and differences, then eliminate them with the necessary identities.
Step-3: Check for angle multiples and use the relevant formulas to eliminate them.
Step-4: If possible, expand any equations, combine like terms, and simplify the equations.
Step-5: Using the Pythagorean identities, replace cos powers higher than 2 with sin powers.
Step-6: Factor numerators and denominators, then cancel any common factors.
Step-7: Both sides should now be exactly equal, or clearly equal, and you should have established your identity.