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What is the value of [(cos 3θ + 2cos 5θ + cos 7θ)÷(cos θ + 2cos 3θ + cos 5θ)] + sin 2θ tan 3θ?
A
cos 2\u03b8
B
sin 2\u03b8
C
tan 2\u03b8
D
cot \u03b8 sin 2\u03b8
Correct Answer:
cos 2\u03b8
Evaluate the following:
$$\frac{{\cos 2\theta \cdot \cos 3\theta - \cos 2\theta \cdot \cos 7\theta + \cos \theta \cdot \cos 10\theta }}{{\sin 4\theta \cdot \sin 3\theta - \sin 2\theta \cdot \sin 5\theta + \sin 4\theta \cdot \sin 7\theta }}$$
A
cot6θ.cot5θ
B
cos6θ.cos5θ
C
cos6θ.cot5θ
D
-cot6θ.cot5θ
The value of $$\frac{{\sec \theta \left( {1 - \sin \theta } \right)\left( {\sin \theta + \cos \theta } \right)\left( {\sec \theta + \tan \theta } \right)}}{{\sin \theta \left( {1 + \tan \theta } \right) + \cos \theta \left( {1 + \cot \theta } \right)}}$$ is equal to:
A
2cosθ
B
cosecθsecθ
C
2sinθ
D
sinθcosθ
If $$\frac{{{{\sin }^2}\theta }}{{{{\cos }^2}\theta - 3\cos \theta + 2}} = 1,\,\theta $$ lies in the first quadrant, then the value of $$\frac{{{{\tan }^2}\frac{\theta }{2} + {{\sin }^2}\frac{\theta }{2}}}{{\tan \theta + \sin \theta }}$$ is:
A
$$\frac{{2\sqrt 3 }}{{27}}$$
B
$$\frac{{7\sqrt 3 }}{{54}}$$
C
$$\frac{{2\sqrt 3 }}{9}$$
D
$$\frac{{5\sqrt 3 }}{{27}}$$
The expression $$\frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta \,{\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}},$$ 0°
A
2tanθ
B
cotθ
C
sinθ
D
2cosθ
The expression $$\frac{{\left( {1 - 2{{\sin }^2}\theta {{\cos }^2}\theta } \right)\left( {\cot \theta + 1} \right)\cos \theta }}{{\left( {{{\sin }^4}\theta + {{\cos }^4}\theta } \right)\left( {1 + \tan \theta } \right){\text{cosec}}\,\theta }} - 1,$$ 0°
A
-sec<sup>2</sup>θ
B
cos<sup>2</sup>θ
C
-sin<sup>2</sup>θ
D
sec<sup>2</sup>θ
$$\frac{{{{\left( {1 + \sec \theta \,{\text{cosec}}\,\theta } \right)}^2}{{\left( {\sec \theta - \tan \theta } \right)}^2}\left( {1 + \sin \theta } \right)}}{{{{\left( {\sin \theta + \sec \theta } \right)}^2} + {{\left( {\cos \theta + {\text{cosec}}\,\theta } \right)}^2}}},$$ 0°
A
1 + sinθ
B
sinθ
C
cosθ
D
1 - cosθ
The value of $$\frac{{2\left( {{{\sin }^6}\theta + {{\cos }^6}\theta } \right) - 3\left( {{{\sin }^4}\theta + {{\cos }^4}\theta } \right)}}{{{{\cos }^4}\theta - {{\sin }^4}\theta - 2{{\cos }^2}\theta }}{\text{ is:}}$$
A
1
B
2
C
-2
D
-1
$$\frac{{1 + \cos \theta - {{\sin }^2}\theta }}{{\sin \theta \left( {1 + \cos \theta } \right)}} \times \frac{{\sqrt {{{\sec }^2}\theta + {\text{cose}}{{\text{c}}^2}\theta } }}{{\tan \theta + \cot \theta }},$$ 0°
A
cosecθ
B
cotθ
C
tanθ
D
secθ
If $$\frac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }} = \frac{1}{7},\,\theta ,$$ lies in first quadrant, then the value of $$\frac{{{\text{cosec}}\,\theta + {{\cot }^2}\theta }}{{{\text{cosec}}\,\theta - {{\cot }^2}\theta }}$$ is:
A
$$\frac{{19}}{5}$$
B
$$\frac{{37}}{{19}}$$
C
$$\frac{{22}}{3}$$
D
$$\frac{{37}}{{12}}$$
Let the function
\[{\text{f}}\left( \theta \right) = \left| {\begin{array}{*{20}{c}} {\sin \theta }&{\cos \theta }&{\tan \theta } \\ {\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)} \\ {\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)} \end{array}} \right|\
A
<br>where \\
B
and \ denote the derivative of f with respect to \. Which of the following statements is/are TRUE?<br>I. There exists \ such that \<br>II. There exists \ such that\
C
<p><span>A.</span> l only
D
</span> ll only