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The question is to find the value of $ \csc \theta + \cot \theta $ if $\sec \theta + \tan \theta = 5$ . Here is what I did : $\sec \theta + \tan \theta = 5$

$\sec \theta = 5 - \tan \theta $

Squaring both sides , $$\sec^2 \theta = 25 + \tan^2 \theta -10\tan \theta$$ Substituting $1+\tan^2 \theta$ for $\sec^2 \theta$ , $$1+\tan^2 \theta = 25 + \tan^2 \theta -10\tan \theta$$ Thus , $$\tan \theta=24/10$$ So , $\cot \theta = 10/24 $ and $\csc \theta=26/24$

Thus $ \csc \theta + \cot \theta =3/2$ . But I checked the answer sheet and the answer is not 3/2 but $(3+\sqrt5 )/2$ . Where have I went wrong ? Please help.

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6  
You went wrong trusting the answer sheet. Your answer is correct. –  Zarrax Aug 10 '13 at 17:49
    
Indeed; if you happened to see it, please disregard my previous erroneous comment. Everything looks good here. –  AWertheim Aug 10 '13 at 17:50
    
Is that really the case ? I'm surprised because the answer sheet is hardly wrong . Is there any online resource I can use to validate answers to questions like this ? –  A Googler Aug 10 '13 at 17:53
    
You can figure out here that a 10-24-26 triangle satisfies the conditions of the question as well as your answer... so you can't be wrong, although on certain questions there may be multiple answers. –  Zarrax Aug 10 '13 at 17:56
1  
Or this followe by this (42 is arbitrary). –  Daniel R Aug 10 '13 at 18:05

2 Answers 2

up vote 8 down vote accepted

Here is a simpler solution to this problem:

$$\left(\sec(\theta)+\tan(\theta) \right)\left(\sec(\theta)-\tan(\theta) \right)=\sec^2(\theta)-\tan^2(\theta)=1$$

Since $\sec(\theta)+\tan(\theta)=5$ you get $\sec(\theta)-\tan(\theta)=\frac{1}{5}$.

Adding and subtracting these two relations you get

$$2\sec(\theta)=5+\frac15=\frac{26}{5} \,;\, 2\tan(\theta)=5-\frac15=\frac{24}{5}$$

Thus $\tan(\theta)=\frac{24}{10}$ and $$\sin(\theta)=\frac{\tan(\theta)}{\sec(\theta)}=\frac{24}{26} \,.$$

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As noted in the comments, you are correct, and the answer key is wrong here.

In cases like these, it's sometimes helpful to check to make sure that you didn't make a mistake in reading the question.

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