Answer:
a) |sec(θ)|
b) 2cot(2x)
Step-by-step explanation:
Trig identities are used to simplify these expressions. Several are used:
- tan(180°-x) = -tan(x)
- cot(90°-x) = tan(x)
- 1 +tan(x)² = sec(x)²
- tan(x) = sin(x)/cos(x)
- sin(2x) = 2sin(x)cos(x)
- cot(x) = cos(x)/sin(x)
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a.
[tex]\sqrt{1-\tan(180^\circ-\theta)\cot(90^\circ-\theta)}=\sqrt{1-(-\tan(\theta))\tan(\theta)}\\\\=\sqrt{1+\tan^2(\theta)}=\sqrt{\sec^2(\theta)}=\boxed{|\sec(\theta)|}[/tex]
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b.
[tex]\dfrac{\cos(2x)\tan(x)}{\sin^2(x)}=\dfrac{\cos(2x)}{\sin^2(x)}\cdot\dfrac{\sin(x)}{\cos(x)}=\dfrac{\cos(2x)}{\sin(x)\cos(x)}\\\\=\dfrac{\cos(2x)}{\left(\dfrac{\sin(2x)}{2}\right)}=2\dfrac{\cos(2x)}{\sin(2x)}=\boxed{2\cot(2x)}[/tex]
