Solve the given trigonometry problem

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The discussion revolves around solving a trigonometry problem involving the relationships between secant and tangent functions. The initial approach simplifies the equation to express x in terms of sine and cosine, leading to the conclusion that x can be represented as sec θ + tan θ. A secondary method suggests comparing x with the identity sec² θ - tan² θ = 1, emphasizing the importance of retaining x in the equation. The conversation highlights different approaches to derive the same trigonometric identity, ultimately confirming the relationship between secant and tangent. The participants appreciate the collaborative effort in exploring various solutions to the problem.
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Homework Statement
If ##x=\sec θ + \tan θ##, then show that ##\dfrac {1}{x}=\sec θ - \tan θ##
Relevant Equations
Trigonometry
My take;

##x^2=\dfrac{(1+\sin θ)^2}{cos^2θ}=\dfrac{(1+\sin θ)^2}{1-\sin ^2θ}=\dfrac{1+\sin θ}{1-\sin θ}##

we know that, ##x=\dfrac{1+\sin θ}{\cos θ}##

##⇒1+\sin θ=x\cos θ##

therefore,

##x^2=\dfrac{x\cos θ}{1-\sin θ}##

##\dfrac{x}{x^2}=\dfrac{1-\sin θ}{\cos θ}##

##\dfrac{1}{x}=\dfrac{1}{\cos θ}-\dfrac{\sin θ}{\cos θ}=\sec θ-\tan θ##

there may be another approach to this problem. Your input highly appreciated...
 
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How about ##x{1\over x} = 1## and comparing that with ##\sec^2 \theta -\tan^2 \theta = 1 ## ?

##\ ##
 
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BvU said:
How about ##x{1\over x} = 1## and comparing that with ##\sec^2 \theta -\tan^2 \theta = 1 ## ?

##\ ##
That is fine, comparing may mean eliminating ##x## from the equation and end up showing that the lhs=rhs in reference to the trig. identities; but that is not what we want. We need ##x## to be part of the problem.
 
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But it is ! If you need it spelled out:$$(\sec + \tan)(\sec-\tan) = x(\sec-tan)=1 \ \& \ x{1\over x}=1 \Rightarrow (\sec-\tan)={1\over x}\\ $$
 
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Ok @BvU ....I will check. Thanks mate.
 
chwala said:
Ok @BvU ....I will check. Thanks mate.
Or ...
A variation, inspired by @BvU :

##\displaystyle x= \sec\theta + \tan\theta##

##\displaystyle x(\sec\theta - \tan\theta)= (\sec\theta + \tan\theta)(\sec\theta - \tan\theta) ##

##\displaystyle \quad \quad \quad \quad \quad \quad \quad =\sec^2\theta - \tan^2\theta ##

##\displaystyle \quad \quad \quad \quad \quad \quad \quad = 1##

Thus: ##\displaystyle \quad \sec\theta - \tan\theta = \dfrac 1 x ##
 
Thanks answer :cool:
use for my ticher
 
or $$ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = \sec^2 \theta - \tan^2 \theta = 1 \implies \sec \theta - \tan \theta = \frac{1}{\sec \theta + \tan \theta} $$
 
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