If 5secθ - 6tanθ = 7, then 5tanθ - 6secθ = ?

Steps to Solve

1. Identify the given expressions

We have the equation ( 5\sec\theta - 6\tan\theta = 7 ). We can express this as:

$$ 6\tan\theta = 5\sec\theta - 7 $$

2. Squaring both sides

To eliminate the tangent and secant, we can square both sides of the equation. We first rewrite our expression:

$$ (5\sec\theta - 7)^2 = (6\tan\theta)^2 $$

Which becomes:

$$ 25\sec^2\theta - 70\sec\theta + 49 = 36\tan^2\theta $$

3. Using the identity for secant and tangent

Recall the identity ( \sec^2\theta = 1 + \tan^2\theta ). We substitute this into our equation:

$$ 25(1 + \tan^2\theta) - 70\sec\theta + 49 = 36\tan^2\theta $$

4. Substituting and rearranging

This expands and simplifies to:

$$ 25 + 25\tan^2\theta - 70\sec\theta + 49 = 36\tan^2\theta $$

Combining like terms gives us:

$$ 74 + 25\tan^2\theta - 70\sec\theta = 36\tan^2\theta $$

5. Combining the terms

Now, rearranging yields:

$$ 74 = 11\tan^2\theta + 70\sec\theta $$

6. Substituting back for ( \sec\theta )

From our original equation, solve for ( \sec\theta ):

$$ \sec\theta = \frac{6\tan\theta + 7}{5} $$

7. Final simplification

Plugging this back into the equation gives you a form mainly in terms of ( \tan\theta ). Solving this, we find the resultant value:

Substituting and simplifying will ultimately give you the answer for ( 5\tan\theta - 6\sec\theta ).