If (sec∂-tan∂)÷(sec∂+tan∂)=36÷49

then find (cosec∂-sec∂)÷(cosec∂+sec∂)

∂=theta

To solve the equation \((\sec \theta - \tan \theta) \div (\sec \theta + \tan \theta) = \frac{36}{49}\), we first need to understand the relationship between secant, tangent, and their respective trigonometric identities. Let's break this down step by step and then use the results to find \((\csc \theta - \sec \theta) \div (\csc \theta + \sec \theta)\).

Step 1: Express secant and tangent in terms of sine and cosine

Recall that:

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

Substituting these definitions into the equation gives:

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

Step 2: Simplifying the expression

This simplifies to:

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

Thus, we can write:

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

Step 3: Cross multiplying

To eliminate the fraction, we can cross-multiply:

Thus, we have:

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

Step 4: Expanding and rearranging

Expanding both sides yields:

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

Now, collect the \(\sin \theta\) terms on one side:

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

Which simplifies to:

13 = 85\sin \theta

Solving for \(\sin \theta\), we find:

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

Step 5: Finding cosecant and secant

Next, we need \(\csc \theta\) and \(\sec \theta\):

• \(\csc \theta = \frac{1}{\sin \theta} = \frac{85}{13}\)
• \(\sec \theta = \frac{1}{\cos \theta}\), and we can find \(\cos \theta\) using \(\sin^2 \theta + \cos^2 \theta = 1\):

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

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

So, \(\cos \theta = \frac{84}{85}\), thus:

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

Step 6: Compute the final expression

Now, we want to find:

\((\csc \theta - \sec \theta) \div (\csc \theta + \sec \theta)\)

Substituting the values we found:

\((\frac{85}{13} - \frac{85}{84}) \div (\frac{85}{13} + \frac{85}{84})\)

Step 7: Simplifying the numerator and denominator

Finding a common denominator for the numerator:

\(\frac{85 \cdot 84}{1092} - \frac{85 \cdot 13}{1092} = \frac{85(84 - 13)}{1092} = \frac{85 \cdot 71}{1092}\)

For the denominator:

\(\frac{85 \cdot 84}{1092} + \frac{85 \cdot 13}{1092} = \frac{85(84 + 13)}{1092} = \frac{85 \cdot 97}{1092}\)

Step 8: Final simplification

Now, putting it all together:

\(\frac{85 \cdot 71 / 1092}{85 \cdot 97 / 1092} = \frac{71}{97}\)

Thus, the final result is:

\((\csc \theta - \sec \theta) \div (\csc \theta + \sec \theta) = \frac{71}{97}\)

Approved