Prove each identity. a) cos x tan^3 x = sin x tan^2 x b) sin^2 θ + cos^4 θ = cos^2 θ + sin^4 θ c) (sin x + cos x)(tan^2 x + 1) / tan x = 1 / cos x + 1 / sin x d) tan^2 β + cos^2 β... Prove each identity. a) cos x tan^3 x = sin x tan^2 x b) sin^2 θ + cos^4 θ = cos^2 θ + sin^4 θ c) (sin x + cos x)(tan^2 x + 1) / tan x = 1 / cos x + 1 / sin x d) tan^2 β + cos^2 β + sin^2 β = 1 / cos^2 β e) sin(π/4 + x) + sin(π/4 - x) = √2 cos x f) sin(π/2 - x) cot(π/2 + x) = -sin x Read more

Steps to Solve

1. Part (a): Proving the First Identity

Start with the left side:

$$ \cos x \tan^3 x = \cos x \left(\frac{\sin^3 x}{\cos^3 x}\right) $$

This simplifies to:

$$ = \frac{\cos x \sin^3 x}{\cos^3 x} = \frac{\sin^3 x}{\cos^2 x} $$

Now, focus on the right side:

$$ \sin x \tan^2 x = \sin x \left(\frac{\sin^2 x}{\cos^2 x}\right) = \frac{\sin^3 x}{\cos^2 x} $$

Both sides are equal, hence proven.

2. Part (b): Proving the Second Identity

Start with the left side:

$$ \sin^2 \theta + \cos^4 \theta $$

Utilize the identity $\cos^2 \theta = 1 - \sin^2 \theta$:

$$ = \sin^2 \theta + (1 - \sin^2 \theta)^2 $$

Expanding the square:

$$ = \sin^2 \theta + (1 - 2\sin^2 \theta + \sin^4 \theta) $$ $$ = 1 - \sin^2 \theta + \sin^4 \theta $$

Now, revert to the right side:

$$ \cos^2 \theta + \sin^4 \theta = (1 - \sin^2 \theta) + \sin^4 \theta $$

Both sides are equal, hence proven.

3. Part (c): Proving the Third Identity

Start with the left side:

$$ ( \sin x + \cos x ) \left( \frac{\tan^2 x + 1}{\tan x} \right) $$

Since $\tan^2 x + 1 = \sec^2 x$:

$$ = ( \sin x + \cos x ) \frac{\sec^2 x}{\tan x} = ( \sin x + \cos x ) \frac{1}{\sin x \cos x} $$

Now, simplify:

$$ = \frac{\sin x + \cos x}{\sin x \cos x} = \frac{1}{\cos x} + \frac{1}{\sin x} $$

Both sides equal, hence proven.

4. Part (d): Proving the Fourth Identity

Consider the left side:

$$ \tan^2 \beta + \cos^2 \beta + \sin^2 \beta $$

Using $\tan^2 \beta = \frac{\sin^2 \beta}{\cos^2 \beta}$ and $ \cos^2 \beta + \sin^2 \beta = 1$:

$$ = \frac{\sin^2 \beta}{\cos^2 \beta} + 1 = \frac{\sin^2 \beta + \cos^2 \beta}{\cos^2 \beta} $$

This leads to:

$$ = \frac{1}{\cos^2 \beta} $$

Right side equals $\sec^2 \beta$, hence proven.

5. Part (e): Proving the Fifth Identity

Consider:

$$ \sin\left(\frac{\pi}{4} + x\right) + \sin\left(\frac{\pi}{4} - x\right) $$

Using the sine addition formula:

$$ = \sqrt{2} \cdot \frac{\sqrt{2}}{2} (\cos x) = \sqrt{2}\cos x $$

Both sides match, hence proven.

6. Part (f): Proving the Sixth Identity

Starting with the left side:

$$ \sin\left(\frac{\pi}{2} - x\right)\cot(x + \frac{\pi}{2}) $$

Using $\sin\left(\frac{\pi}{2} - x\right) = \cos x$ and $\cot\left(x + \frac{\pi}{2}\right) = -\tan x$:

$$ = \cos x (-\tan x) = -\sin x $$

Both sides match, hence proven.