Question
Question: How do you simplify \({{\tan }^{2}}x-{{\cot }^{2}}x\)?...
How do you simplify tan2x−cot2x?
Solution
In this problem we need to simplify the given trigonometric equation. For this we will convert the whole equation in terms of sinx, cosx by using the basic definitions of trigonometric ratios i.e., tanx=cosxsinx, cotx=sinxcosx. Now we will substitute those values in the given equation and simplify them by taking the LCM and calculating the difference. Now we will apply the algebraic formula a2−b2=(a+b)(a−b) and use the trigonometric identities and formulas to simplify the obtained equation. After simplifying the equation, we will get our required solution.
Complete step by step answer:
Given that, tan2x−cot2x.
From the basic definitions of trigonometric ratios substituting the values tanx=cosxsinx, cotx=sinxcosx in the above equation, then we will get
⇒tan2x−cot2x=(cosxsinx)2−(sinxcosx)2
Simplifying the above equation, then we will get
⇒tan2x−cot2x=cos2xsin2x−sin2xcos2x
Taking the LCM and doing the subtraction in the above equation, then we will get
⇒tan2x−cot2x=sin2xcos2xsin2x×sin2x−cos2x×cos2x⇒tan2x−cot2x=(sinxcosx)2(sin2x)2−(cos2x)2
Using the algebraic formula a2−b2=(a+b)(a−b) in the above equation, then we will get
⇒tan2x−cot2x=(sinxcosx)2(sin2x+cos2x)(sin2x−cos2x)
We have the trigonometric identity sin2x+cos2x=1, trigonometric formulas sin2x−cos2x=−cos2x, sinxcosx=2sin2x. Substituting these values in the above equation, then we will get
⇒tan2x−cot2x=(2sin2x)2(1)(−cos2x)
Simplifying the above equation, then we will get
⇒tan2x−cot2x=−4sin22xcos2x⇒tan2x−cot2x=−4sin22xcos2x
Writing the denominator sin22x=sin2x.sin2x, then we will get
⇒tan2x−cot2x=−4×sin2xcos2x×sin2x1
We have the trigonometric formulas sinxcosx=cotx, sinx1=cscx. Applying those formulas in the above equation, then we will get
⇒tan2x−cot2x=−4cot2xcsc2x
Hence the simplified form of the given equation tan2x−cot2x is −4cot2xcsc2x.
Note: In some cases, students may use the trigonometric identities sec2x−tan2x=1, csc2x−cot2x=1 to simplify the equation. From the identity sec2x−tan2x=1, the value of tan2x is sec2x−1 and from the identity csc2x−cot2x=1, the value of cot2x is csc2x−1. Substituting these values in the above equation, then we will get
⇒tan2x−cot2x=(sec2x−1)−(csc2x−1)
Simplifying the above equation, then we will get
⇒tan2x−cot2x=sec2x−1−csc2x+1⇒tan2x−cot2x=sec2x−csc2x
I think the above form is not the simplified form of the given equation. So, I suggest students don’t follow the procedure in this note. Please use the procedure which we discussed in the solution part.