If 7 cosec θ - 3 cot θ = 7 where θ is in first quadrant, then the value of 7 cot θ - 3 cosec θ is equal to :

Question

Accepted Answer

The correct option is 1:

When tackling trigonometric problems, especially those involving relationships between cosecant ($\csc	heta$) and cotangent ($\cot	heta$), the fundamental Pythagorean identity $\csc^2	heta - \cot^2	heta = 1$ is indispensable. This identity can also be written in its difference of squares form: $(\csc	heta - \cot	heta)(\csc	heta + \cot	heta) = 1$. This specific structure often allows for elegant solutions when a linear relationship between $\csc	heta$ and $\cot	heta$ is given, as is the case here.

We are given the equation:
$$7 \csc\theta - 3 \cot\theta = 7 \quad (Equation\ 1)$$

We are asked to find the value of $X$, where:
$$X = 7 \cot\theta - 3 \csc\theta \quad (Equation\ 2)$$

Instead of solving for $\csc	heta$ and $\cot	heta$ individually (which can be cumbersome), we can create a system of equations by cleverly squaring the expressions involving the identity. However, a simpler algebraic manipulation utilizing the difference of squares identity is more efficient for this particular problem structure.

Let's rearrange $Equation\ 1$ to isolate $7\csc\theta$:
$$7 \csc\theta = 7 + 3 \cot\theta$$
Squaring both sides:
$$(7 \csc\theta)^2 = (7 + 3 \cot\theta)^2$$
$$49 \csc^2\theta = 49 + 42 \cot\theta + 9 \cot^2\theta \quad (Equation\ 3)$$

Now, we use the Pythagorean identity $\csc^2\theta = 1 + \cot^2\theta$ to substitute for $\csc^2\theta$ in $Equation\ 3$:
$$49(1 + \cot^2\theta) = 49 + 42 \cot\theta + 9 \cot^2\theta$$
$$49 + 49 \cot^2\theta = 49 + 42 \cot\theta + 9 \cot^2\theta$$

Notice that $49$ cancels out on both sides:
$$49 \cot^2\theta = 42 \cot\theta + 9 \cot^2\theta$$

Rearranging the terms to form a quadratic equation in $\cot\theta$:
$$49 \cot^2\theta - 9 \cot^2\theta - 42 \cot\theta = 0$$
$$40 \cot^2\theta - 42 \cot\theta = 0$$

Factor out $2\cot\theta$:
$$2 \cot\theta (20 \cot\theta - 21) = 0$$

This yields two possible solutions for $\cot	heta$:1.  $2 \cot\theta = 0 \implies \cot\theta = 0$. Since $\theta$ is in the first quadrant, $\cot\theta$ must be positive. Therefore, $\cot\theta = 0$ is not a valid solution for $\theta$ in the first quadrant $(0 \< \theta \< 90^\circ)$.
2.  $20 \cot\theta - 21 = 0 \implies \cot\theta = \frac{21}{20}$. This is the correct value for $\cot\theta$.Now we find $\csc\theta$ using $Equation\ 1$:
$$7 \csc\theta - 3 \left(\frac{21}{20}\right) = 7$$
$$7 \csc\theta = 7 + \frac{63}{20}$$
$$7 \csc\theta = \frac{140 + 63}{20} = \frac{203}{20}$$
$$\csc\theta = \frac{203}{7 \times 20} = \frac{29}{20}$$

Finally, we calculate the value of $X = 7 \cot\theta - 3 \csc\theta$:
$$X = 7 \left(\frac{21}{20}\right) - 3 \left(\frac{29}{20}\right)$$
$$X = \frac{147}{20} - \frac{87}{20}$$
$$X = \frac{147 - 87}{20} = \frac{60}{20} = 3$$

Thus, the value of the expression is 3, corresponding to Option 1.

The incorrect option is 2 (5):
This option suggests an alternative solution that would result from an arithmetic or algebraic error during the calculation. For instance, if one mistakenly used the wrong sign or miscalculated the fractions involved in the final substitution, they might arrive at 5. However, since the trigonometric ratios $\cot\theta = 21/20$ and $\csc\theta = 29/20$ are definitively established by the identity and the given condition, the resulting value must be 3.

The incorrect option is 3 (3/7):
This result is numerically too small and suggests a major error in the substitution or solving process, perhaps mistaking the roles of the coefficients 3 and 7, or dividing by 7 incorrectly at an early stage. It does not align with the solution derived from correctly applying the fundamental trigonometric identity $\csc^2\theta - \cot^2\theta = 1$.

The incorrect option is 4 (5/7):
Similar to Option 3, this value is likely the result of an error in manipulation, such as simplifying the expression before finding the actual values of $\csc\theta$ and $\cot\theta$. No logical mathematical pathway from the given equation $7 \csc\theta - 3 \cot\theta = 7$ leads to $5/7$ as the final answer for $7 \cot\theta - 3 \csc\theta$.