Question

What is the value of \(\theta\)? Statement (I): \(\sec^2\theta=1+\tan^2\theta\) Statement (II): \(\sin\theta+\csc\theta=2\)

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

Whenever an expression contains a trigonometric function and its reciprocal, substitute \(x=\sin\theta\) or \(x=\cos\theta\) to simplify the equation.

Answer 1

The Correct Option is B

Concept: The identity \[ \sec^2\theta=1+\tan^2\theta \] is true for all permissible values of \(\theta\). An identity cannot determine a unique angle.

Step 1: Analyze Statement (I). The given equation is a standard trigonometric identity. It is valid for infinitely many values of \(\theta\). Hence Statement (I) is insufficient.

Step 2: Analyze Statement (II). Given \[ \sin\theta+\csc\theta=2. \] Let \[ x=\sin\theta. \] Then \[ x+\frac1x=2. \] Multiplying by \(x\), \[ x^2+1=2x. \] \[ x^2-2x+1=0. \] \[ (x-1)^2=0. \] \[ x=1. \] Therefore \[ \sin\theta=1. \] Hence \[ \theta=90^\circ \] (modulo full rotations). A unique principal value is obtained. Statement (II) alone is sufficient.