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Question: Solve \(\tan 5x\tan 3x\tan 2x = \) A. \(\tan 5x - \tan 3x - \tan 2x\) B. \((\sin 5x - \sin 3x - ...

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

Explanation

Solution

In order to answer this question, first we will write any of the algebraic equations in which we can include the trigonometric term tan⁡\tan i.e.. 2x=5x−3x2x = 5x - 3x. Then after taking tan⁡\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)=tan⁡a−tan⁡b1+tan⁡a.tan⁡b\tan (a - b) = \dfrac{{\tan a - \tan b}}{{1 + \tan a.\tan b}}

Complete step by step answer:
The given trigonometric expression is: tan⁡5xtan⁡3xtan⁡2x\tan 5x\tan 3x\tan 2x
Now, we can write:
2x=5x−3x2x = 5x - 3x
(when we subtract 3x3x from 5x5x , then we get 2x2x, as we know this simply)
Now, taking tan⁡\tan in both sides of the above equation:
⇒tan⁡2x=tan⁡(5x−3x)\Rightarrow \tan 2x = \tan (5x - 3x)
[as we know the formula: tan⁡(a−b)=tan⁡a−tan⁡b1+tan⁡a.tan⁡b\tan (a - b) = \dfrac{{\tan a - \tan b}}{{1 + \tan a.\tan b}} ]
⇒tan⁡2x=tan⁡5x−tan⁡3x1+tan⁡5xtan⁡3x\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⁡5xtan⁡3x)=tan⁡5x−tan⁡3x ⇒tan⁡2x+tan⁡5xtan⁡3xtan⁡2x=tan⁡5x−tan⁡3x ∴tan⁡5xtan⁡3xtan⁡2x=tan⁡5x−tan⁡3x−tan⁡2x \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⁡5xtan⁡3xtan⁡2x\tan 5x\tan 3x\tan 2x is equal to tan⁡5x−tan⁡3x−tan⁡2x\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.