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