Question

Find the value of \[\tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ \] is
A) 1
B) 2
C) 3
D) None

Answer 1

Hint:
Here, we have to find the value of the product of tangents. We have to use the trigonometric identities to find the value. The tangent of an angle is the trigonometric ratio between the adjacent side and the opposite side of a right triangle containing that angle.

Formula Used:
We will use the following trigonometric identities:
1) \[\tan (90 - \theta ) = \cot \theta \]
2) \[\tan \theta \cdot \cot \theta = 1\]

Complete step by step solution:
We have to find the value of \[\tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ \].
We can rewrite the above equation as:
\[ \Rightarrow \tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ = \tan 5^\circ \tan 10^\circ ...............\tan (90^\circ - 10^\circ )\tan (90^\circ - 5^\circ )\]
Now by using the Trigonometric Identity \[\tan (90 - \theta ) = \cot \theta \], we get
\[ \Rightarrow \tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ = \tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ ...............\cot 10^\circ \cot 5^\circ \]
Now, by rearranging terms, we get
\[ \Rightarrow \tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ = \tan 5^\circ \cot 5^\circ \tan 10^\circ \cot 10^\circ \tan 15^\circ \cot 15^\circ ...............\]
Now, by using the Trigonometric Identity \[\tan \theta \cdot \cot \theta = 1\], we get
\[ \Rightarrow \tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ = 1 \cdot 1 \cdot 1 \cdot 1 \cdot ...............\]
Multiplying the terms on RHS, we get
\[ \Rightarrow \tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ = 1\]
Therefore, the value of \[\tan 5^\circ \tan 10^\circ \tan 15^\circ \tan 20^\circ .............\tan 85^\circ \]is 1.

Hence, option A is the correct answer.

Additional Information:
Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Identity inequalities which are true for every value occurring on both sides of an equation. Geometrically, these identities involve certain functions of one or more angles. There are various distinct identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle. The six basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from the six trigonometric ratios.

Note:
We should know that trigonometric identities are trigonometry equations that are always true, and they're often used to solve trigonometry and geometry problems and understand various mathematical properties. Knowing key trigonometric identities helps you remember and understand important mathematical principles and solve numerous math problems. We should be clear about the correct usage of trigonometric identity at the correct place.