If (sec∂-tan∂)÷(sec∂+tan∂)=36÷49

then find (cosec∂-sec∂)÷(cosec∂+sec∂)

∂=theta

To solve the equation \((\sec \theta - \tan \theta) / (\sec \theta + \tan \theta) = 36 / 49\), we first need to simplify the expression and find the required quantity \((\csc \theta - \sec \theta) / (\csc \theta + \sec \theta)\). Let's break this down step by step.

Understanding the Initial Equation

We start with the equation:

\((\sec \theta - \tan \theta) / (\sec \theta + \tan \theta) = 36 / 49\)

Recall that:

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Substituting these definitions into our expression gives us:

\((\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}) / (\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta})\)

This simplifies to:

\((1 - \sin \theta) / (1 + \sin \theta)\)

Setting Up the Proportions

Now, let's set this equal to \(36 / 49\):

\(\frac{1 - \sin \theta}{1 + \sin \theta} = \frac{36}{49}\)

Cross-multiplying results in:

49(1 - \sin \theta) = 36(1 + \sin \theta)

Distributing gives:

49 - 49\sin \theta = 36 + 36\sin \theta

Now, rearranging terms leads to:

49 - 36 = 49\sin \theta + 36\sin \theta

Thus:

13 = 85\sin \theta

From which we find:

\(\sin \theta = \frac{13}{85}\)

Finding Cosecant and Secant Values

Next, we need to calculate \((\csc \theta - \sec \theta) / (\csc \theta + \sec \theta)\). First, we need to find \(\csc \theta\) and \(\sec \theta\):

\(\csc \theta = \frac{1}{\sin \theta} = \frac{85}{13}\)

\(\sec \theta = \frac{1}{\cos \theta}\)

To find \(\cos \theta\), we can use the Pythagorean identity:

\(\sin^2 \theta + \cos^2 \theta = 1\)

Calculating \(\cos^2 \theta\):

\(\cos^2 \theta = 1 - \left(\frac{13}{85}\right)^2 = 1 - \frac{169}{7225} = \frac{7225 - 169}{7225} = \frac{7056}{7225}\)

Thus, \(\cos \theta = \sqrt{\frac{7056}{7225}} = \frac{84}{85}\).

Now we can find \(\sec \theta\):

\(\sec \theta = \frac{1}{\cos \theta} = \frac{85}{84}\)

Putting It All Together

Now substituting \(\csc \theta\) and \(\sec \theta\) into the expression:

\(\frac{\csc \theta - \sec \theta}{\csc \theta + \sec \theta} = \frac{\frac{85}{13} - \frac{85}{84}}{\frac{85}{13} + \frac{85}{84}}\)

Finding a common denominator for the numerators and denominators:

The common denominator of 13 and 84 is 1092. Thus:

Numerator: \(\frac{85 \times 84 - 85 \times 13}{1092} = \frac{7140 - 1105}{1092} = \frac{6035}{1092}\)

Denominator: \(\frac{85 \times 84 + 85 \times 13}{1092} = \frac{7140 + 1105}{1092} = \frac{8245}{1092}\)

Finally, simplifying gives:

\(\frac{6035}{8245}\), which can be reduced to \(\frac{13}{17}\) after finding the GCD.

The Final Answer

Therefore, the value of \((\csc \theta - \sec \theta) / (\csc \theta + \sec \theta)\) is:

\(\frac{13}{17}\)