Question

What is the value of \[\tan 25^\circ \tan 15^\circ + \tan 15^\circ \tan 50^\circ + \tan 25^\circ \tan 50^\circ \] ?
(a). 0
(b). 1
(c). 2
(d). 4

Answer 1

Hint: Take tan 15° as a common term and use tangent of the sum of angles, that is, \[\tan (x + y) = \dfrac{{\tan x + \tan y}}{{1 - \tan x\tan y}}\] to simplify the given trigonometric function.

Complete step-by-step answer:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle is called trigonometric functions.

The sine, cosine, tangent, cotangent, secant, and cosecant are the trigonometric functions.
The sine and cosecant are inverses of each other. The cosine and secant are inverses of each other. The tangent and the cotangent are inverses of each other.

They all are related to one another with special formulas.

We know that the tangent of the sum of two angles is given by the formula as follows:

\[\tan (x + y) = \dfrac{{\tan x + \tan y}}{{1 - \tan x\tan y}}\]

From this formula, we have:

\[\tan x + \tan y = \tan (x + y)(1 - \tan x\tan y)........(1)\]

Let us assign the value of \[\tan 25^\circ \tan 15^\circ + \tan 15^\circ \tan 50^\circ + \tan

25^\circ \tan 50^\circ \] to the variable T.

\[T = \tan 25^\circ \tan 15^\circ + \tan 15^\circ \tan 50^\circ + \tan 25^\circ \tan 50^\circ \]

Taking tan 15° as a common term, we have:

\[T = \tan 15^\circ (\tan 25^\circ + \tan 50^\circ ) + \tan 25^\circ \tan 50^\circ ........(2)\]

Applying formula (1) to equation (2), we have the following:

\[T = \tan 15^\circ \tan (25^\circ + 50^\circ )(1 - \tan 25^\circ \tan 50^\circ ) + \tan 25^\circ

\tan 50^\circ \]

Simplifying, we have:

\[T = \tan 15^\circ \tan 75^\circ (1 - \tan 25^\circ \tan 50^\circ ) + \tan 25^\circ \tan 50^\circ

\]

We know that \[\tan x = \cot (90^\circ - x)\], then tan 15° is equal to cot (90° - 15°) which is

 cot 75°.

\[T = \cot 75^\circ \tan 75^\circ (1 - \tan 25^\circ \tan 50^\circ ) + \tan 25^\circ \tan 50^\circ

\]

We know that \[\tan x = \dfrac{1}{{\cot x}}\], then we have \[\tan x\cot x = 1\]. Hence, we

have as follows:

\[T = 1(1 - \tan 25^\circ \tan 50^\circ ) + \tan 25^\circ \tan 50^\circ \]

Simplifying the terms in the bracket, we get:

\[T = 1 - \tan 25^\circ \tan 50^\circ + \tan 25^\circ \tan 50^\circ \]

Canceling the term \[\tan 25^\circ \tan 50^\circ \], we have:

\[T = 1\]

Hence, the correct answer is option (b).

Note: You can equally take tan25° as the common term or tan50° as the common term and simplify to get the final answer.