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If sin θ sec2 θ = 2/3, 0° < θ < 90°, then the value of (tan2 θ + sin2 θ) is:
1. 7/12
2. 13/12
3. 11/12
4. 5/4

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Correct Answer - Option 1 : 7/12

Given:

sin θ sec2 θ = 2/3

Concept used:

Value putting method.

Calculation:

sinθ × sec2θ = 2/3 where θ° < θ < 90° 

Let us take θ = 30° 

sin30° = 1/2

sec30° = \(\frac{2}{{\sqrt 3 }}\)

sin30° × sec230° = \(\frac{1}{2} × {\left( {\frac{2}{{\sqrt 3 }}} \right)^2}\)

sin30° × sec230° = \(\frac{1}{2} \times \frac{4}{3} = \frac{2}{3}\)

Hence proved.

Now, (tan2θ + sin2θ) = (tan230° + sin230°)

(tan230° + sin230°) = \(\left( {{{\left( {\frac{1}{{\sqrt 3 }}} \right)}^2} + {{\left( {\frac{1}{2}} \right)}^2}} \right)\)

(tan230° + sin230°) = \(\frac{1}{3} + \frac{1}{4}\)

(tan230° + sin230°) = 7/12

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