Question

If sec theta = x+1/4x ,prove that :
sec theta +tan theta=2x or 1/2x.

Answer 1

Answer:

Hope this will help you

Answer 2

Answer:

If sec theta = x+1/4x then sec theta +tan theta=2x or 1/2x.

Step-by-step explanation:

$Given \: sec\theta = x + \frac{1}{4x}--(1)$

/* We know the, trigonometric identity :

tan²A = Sec²A-1*/

$tan^{2}\theta \\= \left(x+\frac{1}{4x}\right)^{2}-1\\=x^{2}+2\times x \times \big(\frac{1}{4x}\big)+\left(\frac{1}{4x}\right)^{2}-1$

$=x^{2}+\frac{1}{2}+\frac{1}{16x^{2}}-1$

$=x^{2}-\frac{1}{2}+\frac{1}{16x^{2}}$

$=x^{2}-2\times x \times \frac{1}{4x}+\big(\frac{1}{4x}\big)^{2}$

$=\left(x-\frac{1}{4x}\right)^{2}$

$tan\theta = ± \left(x-\frac{1}{4x}\right)--(2)$

$Now,\\sec\theta + tan\theta\\=x+\frac{1}{4x}±\left(x-\frac{1}{4x}\right)$

$Case \:1\\sec\theta + tan\theta\\=x+\frac{1}{4x}+\left(x-\frac{1}{4x}\right)$

$=2x$

$Case\:2\\sec\theta + tan\theta\\=x+\frac{1}{4x}-\left(x-\frac{1}{4x}\right)$

$=x+\frac{1}{4x}-x+\frac{1}{4x}$

$=\frac{1}{2x}$

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