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What is sec2xtan2x in terms of tan ?

Answer
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Hint: In this question, we need to convert sec2xtan2x in the terms of tangent . To convert sec2xtan2x in terms of tangent expression , we will use the Trigonometric identities and functions. The basic trigonometric functions are sine , cosine and tangent. In trigonometry , the tangent function is used to find the slope of a line. Also with the help of algebraic formulae, we can easily convert in the terms of tangent.
Identity used :
sin2θ + cos2θ=1

Formula used :
1. cos 2θ =cos2θsin2θ
2. sin 2θ = 2 sin θ cos θ
3. tanA tanB1 +tanA×tanB=tan(AB)
Algebraic formulae used :
1. a2+b22ab=(a+b)2
2. a2b2=(a+b)(ab)

Complete step-by-step solution:
Given,
sec2xtan2x
We need to convert the given expression in terms of tangent .
We know that sec θ=1cos θ and also tan θ=sin θcos θ
Thus we get,
sec2xtan2x=(1cos2x)(sin2xcos2x)
(1sin2x)cos2x
By applying the formula,
We get,
(12sinxcosx)cos2xsin2x 
By using the identity , We can substitute
sin2x +cos2x  in the place of 1
(sin2x +cos2x 2sinx2cosx)cos2xsin2x 
We know that a2+b22ab=(a+b)2
Thus we can write
sin2x+cos2x2sinxcosx  as (cos x sin x)2 (since 0 < x <π4 then sin x < cos x )
(cos x sin x)2 cos2xsin2x 
We know that
a2b2=(a+b)(ab)
Thus we can write
cos2xsin2x=(cos x+ sin x)(cos x sin x)
(cos x sin x)2(cosx+sin x)(cos x sin x) 
By simplifying,
We get,
cos x  sin xcos x + sin x
By taking cosx outside from both numerator and denominator,
We get,
cos x[1  (sinxcos x)](cos x[1 + (sinxcos x)])
On simplifying,
We get,
1 tanx1 +tanx
We can write this expression as
1 tanx1 +1×tanx in order to bring the expression in the form of tan(A+B) formula.
We know that the value of tan(π4) is 1
tan(π4) tanx1 +tan(π4)×tanx
We know
tan A tanB1 +tanA×tanB=tan(AB)
By applying the formula we get ,
tan(π4)tanx1 +tan(π4)×tanx=tan((π4)x)
Thus we get,
sec2xtan2x=tan((π4)x)
Therefore we have converted the given expression in terms of tangent.
Final answer :
sec2xtan2x in terms of tan is tan((π4)x)


Note: The concept used to solve the given problem is trigonometric identities and ratios. Trigonometric identities are nothing but they involve trigonometric functions including variables and constants. The common technique used in this problem is the use of algebraic formulae with the use of trigonometric functions.

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