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# If cos2$$\theta$$ + $$\frac{1}{8}$$ = sin2$$\theta$$, find tan$$\theta$$...

### Question

If cos2$$\theta$$ + $$\frac{1}{8}$$ = sin2$$\theta$$, find tan$$\theta$$

### Options

A) 3

B) $$\frac{3\sqrt{7}}{7}$$

C) 3$$\sqrt{7}$$

D) $$\sqrt{7}$$

### Explanation:

cos2$$\theta$$ + $$\frac{1}{8}$$ = sin2$$\theta$$..........(i)
from trigometric ratios for an acute angle, where cos$$\theta$$ + sin2$$\theta$$ = 1 - cos$$\theta$$ ........(ii)
Substitute for equation (i) in (i) = cos2$$\theta$$ + $$\frac{1}{8}$$ = 1 - cos2$$\theta$$
= cos2$$\theta$$ + cos2$$\theta$$ = 1 - $$\frac{1}{8}$$
2 cos2$$\theta$$ = $$\frac{7}{8}$$
cos2$$\theta$$ = $$\frac{7}{2 \times 3}$$
$$\frac{7}{16}$$ = cos$$\theta$$
$$\sqrt{\frac{7}{16}}$$ = $$\frac{\sqrt{7}}{4}$$
but cos $$\theta$$ = $$\frac{\text{adj}}{\text{hyp}}$$
opp2 = 42 ($$\sqrt{7}$$)2
= 16 - 7
opp = $$\sqrt{9}$$ = 3
than $$\theta$$ = $$\frac{\text{opp}}{\text{hyp}}$$
= $$\frac{3}{\sqrt{7}}$$
$$\frac{3}{\sqrt{7}}$$ x $$\frac{7}{\sqrt{7}}$$ = $$\frac{3\sqrt{7}}{7}$$

## Dicussion (1)

• cos2$$\theta$$ + $$\frac{1}{8}$$ = sin2$$\theta$$..........(i)
from trigometric ratios for an acute angle, where cos$$\theta$$ + sin2$$\theta$$ = 1 - cos$$\theta$$ ........(ii)
Substitute for equation (i) in (i) = cos2$$\theta$$ + $$\frac{1}{8}$$ = 1 - cos2$$\theta$$
= cos2$$\theta$$ + cos2$$\theta$$ = 1 - $$\frac{1}{8}$$
2 cos2$$\theta$$ = $$\frac{7}{8}$$
cos2$$\theta$$ = $$\frac{7}{2 \times 3}$$
$$\frac{7}{16}$$ = cos$$\theta$$
$$\sqrt{\frac{7}{16}}$$ = $$\frac{\sqrt{7}}{4}$$
but cos $$\theta$$ = $$\frac{\text{adj}}{\text{hyp}}$$
opp2 = 42 ($$\sqrt{7}$$)2
= 16 - 7
opp = $$\sqrt{9}$$ = 3
than $$\theta$$ = $$\frac{\text{opp}}{\text{hyp}}$$
= $$\frac{3}{\sqrt{7}}$$
$$\frac{3}{\sqrt{7}}$$ x $$\frac{7}{\sqrt{7}}$$ = $$\frac{3\sqrt{7}}{7}$$