Question

if 5 tan theta =4 , write the value of (cos theta -sin theta / cos theta +sin theta) .​

Answer 1

Given:-

• 5 tanθ = 4

To Find:-

• The value of:-

• $\sf{\dfrac{Cos\theta - Sin\theta}{Cos\theta + Sin\theta}}$

Solution:-

We have:-

• 5 tanθ = 4

⇒ tanθ = 4/5

We know,

• $\boxed{\sf{Tan\theta = \dfrac{Sin\theta}{Cos\theta}}}$

Hence we can write tanθ = 4/5 as:-

$\sf{\dfrac{Sin\theta}{Cos\theta} = \dfrac{4}{5}}$

On Comparing we get:-

• Sinθ = 4
• Cosθ = 5

Now,

We need to find the value of:-

$\sf{\dfrac{Cos\theta - Sin\theta}{Cos\theta + Sin\theta}}$ . . . . . . . . . . (i)

Putting the value of Sinθ and Cosθ in (i)

= $\sf{\dfrac{5-4}{5+4}}$

= $\sf{\dfrac{1}{9}}$

∴ The value of $\sf{\dfrac{Cos\theta - Sin\theta}{Cos\theta + Sin\theta} = \dfrac{1}{9}}$

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Some other things to be known:-

$\boxed{\begin{array}{cc} \sf{Sin\theta = \dfrac{1}{cosec\theta}} & \sf{Cosec\theta = \dfrac{1}{Sin\theta}} \\\\ \sf{Cos\theta = \dfrac{1}{Sec\theta}} & \sf{Sec\theta = \dfrac{1}{Cos\theta}} \\\\ \sf{Tan\theta = \dfrac{1}{Cot\theta}} & \sf{Cot\theta = \dfrac{1}{Tan\theta}} \end{array}}$

Also:-

$\boxed{\begin{array} {cc} \sf{Tan\theta = \dfrac{Sin\theta}{Cos\theta}} & \sf{Cot\theta = \dfrac{Cos\theta}{Sin\theta}}\end{array}}$

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