Question

if sec theta = 5/ 4 verify that tan theta / 1+ tan square theta = sin theta / sec theta.​

Answer 1

Step-by-step explanation:

Given- Sec theta= 5/4

hypotenuse = 5

adjacent side = 4

Then by Pythagoras theorem-

oppsite side = root ((5)^2 - (4)^2)

= root (25 - 16)

= root 9

opposite side = 3

tan theta = 3/4

sin theta = 3/5

(3 \div 4 )\div 1 +( 3 \div 4) ^{2}(3÷4)÷1+(3÷4)

2

3/4 ÷ 1 + 9/16

3/4 ÷ 25/16

= 12/25 = LHS

3/5 ÷ 5/4

= 12/25 = RHS

LHS = RHS

Hence Proved

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Answer 2

Step-by-step explanation:

$\sf\large\underline\purple{Given:-}$

Here,

$\displaystyle \star \: \sin( \theta) = \frac{5}{4}$

$\sf\large\underline\purple{Solution:-}$

$\displaystyle \sf \tt \implies \: \frac{\tan( \theta)}{1 + \tan( \theta) } = \frac{ \sin( \theta) }{ \sec( \theta ) }$

By using LHS,

$\displaystyle \star \sf \: \sin( \theta) = \frac{p}{h} = \frac{5}{4} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \displaystyle \sf \star \: then \: b \: = 3 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \displaystyle \star \sf \: \frac{ \tan( \theta) }{1 + \tan( \theta) } = \frac{ \frac{3}{4} }{ \frac{16 + 9}{16} } = \frac{12}{25}$

By using RHS,

$\displaystyle \sf \star \: \sin( \theta) . \ \cos( \theta) = \frac{3}{5} \times \frac{4}{5} = \frac{12}{25}$

Hence LHS = RHS

Verified