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Question: The value of \(\tan \theta \left( {1 - {{\cot }^2}\theta } \right)\) is equal to (a). \(\cot \thet...

The value of tan⁡θ(1−cot⁡2θ)\tan \theta \left( {1 - {{\cot }^2}\theta } \right) is equal to
(a). cot⁡θ(1−tan⁡2θ)\cot \theta \left( {1 - {{\tan }^2}\theta } \right)
(b). cot⁡θ(tan⁡2θ−1)\cot \theta \left( {{{\tan }^2}\theta - 1} \right)
(c). cot⁡θtan⁡2θ\cot \theta {\tan ^2}\theta
(d). tan⁡θcos⁡ec2θ\tan \theta \cos e{c^2}\theta

Explanation

Solution

Here, we will use a trigonometric relation between tan⁡θ\tan \theta and cot⁡θ\cot \theta to solve the problem. This kind of problem is generally solved by seeing the relationship between the problem statement and the answer options to proceed step by step.

Complete step by step solution:
tan⁡θ(1−cot⁡2θ)\tan \theta \left( {1 - {{\cot }^2}\theta } \right) (This is the problem which we have to solve using trigonometric relation)

By opening the bracket, we will get:
= tan⁡θ−tan⁡θ×cot⁡2θ...........(1)\tan \theta - \tan \theta \times {\cot ^2}\theta ...........\left( 1 \right)
(Let us consider this as equation 1)
Since, trigonometric relation between tan⁡θ\tan \theta and cot⁡θ\cot \theta is: cot⁡=1tan⁡θ=1tan⁡θ\cot = \frac{1}{{\tan \theta }} = \frac{1}{{\tan \theta }} and cot⁡2θ=1tan⁡2θ{\cot ^2}\theta = \frac{1}{{{{\tan }^2}\theta }}

By substituting the value cot⁡2θ{\cot ^2}\theta , we can convert the equation (1) into:
= tan⁡θ−tan⁡θ×1tan⁡2θ\tan \theta - \tan \theta \times \frac{1}{{{{\tan }^2}\theta }}
= tan⁡θ−1tan⁡θ\tan \theta - \frac{1}{{\tan \theta }}
We can see that now we only get the equation in tan⁡θ\tan \theta

By solving further, we will get:
= tan⁡2θ−1tan⁡θ\frac{{{{\tan }^2}\theta - 1}}{{\tan \theta }}
Now, by taking 1tan⁡θ\frac{1}{{\tan \theta }}as common, we will get:
= 1tan⁡θ(tan⁡2θ−1)\frac{1}{{\tan \theta }}\left( {{{\tan }^2}\theta - 1} \right)
Since, we know that 1tan⁡θ=cot⁡θ\frac{1}{{\tan \theta }} = \cot \theta , we can convert it into:
= cot⁡θ(tan⁡2θ−1)\cot \theta \left( {{{\tan }^2}\theta - 1} \right)

Hence, option (b) is the correct answer.

Note: In this kind of question, it is important to look at the answer options while looking towards the question and to proceed accordingly step by step. We can try to solve and match one of the options steps by step. For solving such a problem, we need to remember the trigonometric relations.