If 4 cosec θ + 5 cot θ = 7, then cosec θ + 4 cot θ = ?

Steps to Solve

1. Define variables for simplicity

Let ( x = \csc(\theta) ) and ( y = \cot(\theta) ). So the equation ( 4 \csc(\theta) + 5 \cot(\theta) = 7 ) can be rewritten as: $$ 4x + 5y = 7 $$

2. Solve for one variable

From the equation ( 4x + 5y = 7 ), isolate ( y ): $$ 5y = 7 - 4x $$ $$ y = \frac{7 - 4x}{5} $$

3. Substitute and simplify

Now, we want to find the expression ( x + 4y ). Substitute ( y ): $$ x + 4y = x + 4\left(\frac{7 - 4x}{5}\right) $$ Combine the terms: $$ x + \frac{28 - 16x}{5} $$

4. Combine the fractions

To combine the terms into a single fraction, use a common denominator of 5: $$ \frac{5x + 28 - 16x}{5} $$ This simplifies to: $$ \frac{-11x + 28}{5} $$

5. Find the value using the original equation

To solve for ( x ), we can use the original equation ( 4x + 5y = 7 ). Since we can choose ( y ) in terms of ( x ) and back substitute later for ( x ), we can try values for ( x ) that satisfy both equations, or directly substitute reasonable values from the range of trigonometric functions.

Assuming ( y = 0 ) (which works at some point since these terms can reach ( 4 )): $$ 4x + 5(0) = 7 $$ $$ 4x = 7 $$ $$ x = \frac{7}{4} $$

Now substitute ( x ) back in: $$ \frac{-11\left(\frac{7}{4}\right) + 28}{5} = \frac{-\frac{77}{4} + 28}{5} = \frac{-\frac{77}{4} + \frac{112}{4}}{5} = \frac{\frac{35}{4}}{5} = \frac{35}{20} = \frac{7}{4} $$

Thus, ( \csc(\theta) + 4 \cot(\theta) = \frac{7}{4} ).