1.8 E: Trigonometry

Prove that

\tan \phi + \cot \phi \equiv 2 \csc 2\phi

for $ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}$ .

[3]

Using the identity in part (a), or otherwise, prove that

\cot^2 \phi - \tan^2 \phi \equiv 4 \cot 2\phi \csc 2\phi

for $ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}$ .

[3]

Hence solve, for $−π2<α<π2-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$ ,

\cot 2\alpha \csc 2\alpha = 3 \tan^2 \alpha + 1

giving your answers to 2 decimal places.

[4]

Prove that

\tan \phi + \cot \phi \equiv 2 \csc 2\phi

for $ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}$ .

[3]

Using the identity in part (a), or otherwise, prove that

\cot^2 \phi - \tan^2 \phi \equiv 4 \cot 2\phi \csc 2\phi

for $ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}$ .

Hence solve, for $−π2<α<π2-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$ ,

\cot 2\alpha \csc 2\alpha = 3 \tan^2 \alpha + 1

giving your answers to 2 decimal places.

[4]

1.8 E: Trigonometry Questions