Question

Sectheta + tantheta = p , show that sectheta - tantheta = 1/p. Hence , find the values of costheta and sintheta

Answer 1

Answer:

$\red { cos\theta } \green { = \frac{2p}{p^{2}+1}}$

$\red{sin\theta} \green {= \frac{p^{2} + 1 }{p^{2} - 1 }}$

Step-by-step explanation:

$Given \: sec\theta + tan\theta = p \:---(1)$

$We \: know \: the \: Trigonometric \: identity$

$\boxed { \pink { sec^{2}\theta - tan^{2}\theta = 1 }}$

$\implies (sec\theta+tan\theta)(sec\theta - tan\theta) = 1$

$\implies p (sec\theta - tan\theta) = 1$

$\implies sec\theta - tan\theta = \frac{1}{p}\:--(2)$

/* Add equation (1) and (2) , we get

$\implies 2sec\theta = p + \frac{1}{p}$

$\implies \frac{2}{cos\theta} = \frac{p^{2} + 1 }{p}$

$\implies cos\theta = \frac{2p}{p^{2}+1} \:--(3)$

/* Subtract equation (2) from (1) , we get

$\implies 2tan\theta = p - \frac{1}{p}$

$\implies \frac{2sin\theta}{cos\theta} = \frac{p^{2} - 1 }{p}$

$\implies \frac{sin\theta}{cos\theta}= \frac{2p}{p^{2}-1} \:--(4)$

/* Do (4) ÷ (3) , we get

$\implies sin\theta = \frac{p^{2} + 1 }{p^{2} -1 }\:--(4)$

Therefore.,

$\red { cos\theta } \green { = \frac{2p}{p^{2}+1}}$

$\red{sin\theta} \green {= \frac{p^{2} + 1 }{p^{2} - 1 }}$

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