Question

If we have the expression \[A+C=B,\]then \[\tan A\tan B\tan C=\]
 1). \[\tan A\tan B\tan C\]
2). \[\tan B-\tan C-\tan A\]
 3). \[\tan A+\tan C-\tan B\]
 4). \[-(\tan A\tan B\tan C)\]

Answer 1

Hint: To solve this question use the concept of trigonometry and use the formula for the given question. We can easily find the value of a given question. Check all the options one by one by using the concept of trigonometry and calculation. After checking all the options you can get the correct options from the given options.

Complete step-by-step solution:
For solving this question, we should have the knowledge of trigonometry and its properties ,here we have to use the properties of basic trigonometry.
As we know that sum of all angles in any triangle is supplementary but here in this condition of angle is given so we will use that condition.
Some basic trigonometric properties are that cotangent is reciprocal of tangent , Cosecant is reciprocal of Sine and Secant is reciprocal of cosine. Tangent is also expressed as the ratio of Sine and Cosine function.
In this above question, we have given the equation
\[A+C=B\] .
By solving this we can get the value of the given expression which we have to calculate.
As, we know we have a given condition for angles is ,
\[A+C=B\]
 so, we multiply both sides by tangent function then mathematically we can represent this above expression as,
\[\tan (A+C)=\tan B\] ……. (1)
We know that, the formula for finding the value of
\[\tan (a+b)=\dfrac{\tan a+\tan b}{1-\tan a\tan b}\] ……..(2)
 So apply this formula on left hand side in equation (1) we get ,
\[\tan (A+C)\]=\[\dfrac{\tan A+\tan C}{1-\tan A\tan C}\] …..…. (3)
Substituting the value of equation (1) in equation (3) in place of \[\tan (A+C)\], we can express above expression as ,
 \[\Rightarrow \]\[\dfrac{\tan A+\tan C}{1-\tan A\tan C}\]=\[\tan B\]
Here, we will do the cross multiplication of above expression, we get \[\Rightarrow \]\[\tan A+\tan C\]= \[\tan B\](\[1-\tan A\tan C\]) …….. (4)
Now, we simplify the above equation (4),we get
\[\Rightarrow \]\[\tan A+\tan C\]=\[\tan B\]-\[\tan A\tan B\tan C\] …. (5)
 Now arranging these terms we can simplify above expression as,
\[\Rightarrow \]\[\tan A\tan B\tan C\]=\[\tan B-\tan C-\tan A\]
So we get the required value of the given expression.
Hence, we can conclude that the correct option is (2).

Note: Trigonometry is the study of right angle triangles and the lengths and angles of the triangles and it is the most important field of. There are six trigonometric ratios like Sine , Cosine , Tangent , Cotangent , secant and cosecant .These are generally used to find angles and their sides and these can be expressed in terms of the sides of a right-angled triangle for a specific angle\[\theta \].