Question

If \[ \sin\theta+\cosec\theta=4, \] then \[ \sin^2\theta+\cosec^2\theta= \]

• A. \(12\)
• B. \(18\)
• C. \(16\)
• D. \(14\)

Hint:

Whenever expressions of the form \[ x+\frac1x \] are given, square both sides to obtain \[ x^2+\frac1{x^2} \] using \[ \left(x+\frac1x\right)^2=x^2+\frac1{x^2}+2. \]

Answer 1

The Correct Option is D

Step 1: Write the given equation.
We are given $\sin\theta + \csc\theta = 4$.
Step 2: Square both sides.
$(\sin\theta + \csc\theta)^2 = 16$.
Step 3: Expand the square.
$\sin^2\theta + 2\sin\theta\csc\theta + \csc^2\theta = 16$.
Step 4: Simplify using the identity $\sin\theta \cdot \csc\theta = 1$.
The middle term $2\sin\theta\csc\theta = 2 \times 1 = 2$. So the equation becomes: $\sin^2\theta + \csc^2\theta + 2 = 16$.
Step 5: Solve for the required expression.
$\sin^2\theta + \csc^2\theta = 16 - 2 = 14$.
Step 6: Match with the options.
The value is 14, which corresponds to option (4).
\[ \boxed{14} \]