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Question: Prove that \(\dfrac{{1 - {{\tan }^2}\theta }}{{{{\cot }^2}\theta - 1}} = {\tan ^2}\theta \)....

Prove that 1−tan⁡2θcot⁡2θ−1=tan⁡2θ\dfrac{{1 - {{\tan }^2}\theta }}{{{{\cot }^2}\theta - 1}} = {\tan ^2}\theta .

Explanation

Solution

We will put in the formula tan⁡θ=sin⁡θcos⁡θ\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }} in the LHS of the question and then simplify it, and then put in sin⁡θcos⁡θ=tan⁡θ\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta again and match it to the RHS and thus, we are sorted.

Complete step-by-step answer:
Let us consider the LHS that is: 1−tan⁡2θcot⁡2θ−1\dfrac{{1 - {{\tan }^2}\theta }}{{{{\cot }^2}\theta - 1}}.
Now put in tan⁡θ=sin⁡θcos⁡θ\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }} and cot⁡θ=1tan⁡θ\cot \theta = \dfrac{1}{{\tan \theta }} in the LHS, we will have with us the following expression:-
LHS = 1−(sin⁡θcos⁡θ)21tan⁡2θ−1\dfrac{{1 - {{\left( {\dfrac{{\sin \theta }}{{\cos \theta }}} \right)}^2}}}{{\dfrac{1}{{{{\tan }^2}\theta }} - 1}}
Now put in tan⁡θ=sin⁡θcos⁡θ\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }} again in the above expression, we will get:-
LHS = 1−sin⁡2θcos⁡2θcos⁡2θsin⁡2θ−1\dfrac{{1 - \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}}}{{\dfrac{{{{\cos }^2}\theta }}{{{{\sin }^2}\theta }} - 1}}
On simplifying it, we will get:-
LHS = cos⁡2θ−sin⁡2θcos⁡2θcos⁡2θ−sin⁡2θsin⁡2θ\dfrac{{\dfrac{{{{\cos }^2}\theta - {{\sin }^2}\theta }}{{{{\cos }^2}\theta }}}}{{\dfrac{{{{\cos }^2}\theta - {{\sin }^2}\theta }}{{{{\sin }^2}\theta }}}}
We can rewrite it as:-
LHS = cos⁡2θ−sin⁡2θcos⁡2θ×sin⁡2θcos⁡2θ−sin⁡2θ\dfrac{{{{\cos }^2}\theta - {{\sin }^2}\theta }}{{{{\cos }^2}\theta }} \times \dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta - {{\sin }^2}\theta }}
Simplifying it further, we will get:-
LHS = sin⁡2θcos⁡2θ\dfrac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta }}
Now putting in sin⁡θcos⁡θ=tan⁡θ\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta , we will get:-
LHS = tan⁡2θ{\tan ^2}\theta that is equal to RHS.
Hence, proved.

Note: The students must note that to tackle the problem with tangent and cotangent. They must always convert them to sine and cosine.
Trigonometry (from the Greek trigonon = three angles and metron = measure) is a part of elementary mathematics dealing with angles, triangles and trigonometric functions such as sine (abbreviated sin), cosine (abbreviated cos) and tangent (abbreviated tan). It has some connection to geometry, although there is disagreement on exactly what that connection is; for some, trigonometry is just a section of geometry.
Fun Facts about Trigonometry:-
The word "Trigonometry" comes from the word "Triangle Measure".
There are 8 Trigonometric identities called fundamental identities. 3 of them are called Pythagorean identities as they are based on Pythagorean Theorem.
The field emerged from applications of Geometry to astronomical studies in the 3rd Century BC.
Trigonometry is associated with music and architecture.
Engineers use Trigonometry to figure out the angles of the sound waves and how to design rooms.