TanX + secX ×tanX - secX =2(tanX. CosecX- Cox.secX

Question

Simplify

 

Answer 1

Let's simplify the given trigonometric expression step by step:

Given expression: \( \tan(x) + \sec(x) \cdot \tan(x) - \sec(x) = 2(\tan(x) \cdot \csc(x) - \cot(x) \cdot \sec(x)) \)

Step 1: Factor out a common factor of \( \tan(x) \) on the left side:

\( \tan(x) \cdot (1 + \sec(x)) - \sec(x) = 2(\tan(x) \cdot \csc(x) - \cot(x) \cdot \sec(x)) \)

Step 2: Rewrite \( \sec(x) \) as \( \frac{1}{\cos(x)} \):

\( \tan(x) \cdot (1 + \frac{1}{\cos(x)}) - \frac{1}{\cos(x)} = 2(\tan(x) \cdot \csc(x) - \cot(x) \cdot \sec(x)) \)

Step 3: Find a common denominator on the left side:

\( \frac{\tan(x) \cdot (\cos(x) + 1) - 1}{\cos(x)} = 2(\tan(x) \cdot \csc(x) - \cot(x) \cdot \sec(x)) \)

Step 4: Multiply both sides by \( \cos(x) \) to clear the fraction:

\( \tan(x) \cdot (\cos(x) + 1) - 1 = 2(\tan(x) \cdot \csc(x) - \cot(x) \cdot \sec(x)) \)

Step 5: Distribute on both sides:

\( \tan(x) \cdot \cos(x) + \tan(x) - 1 = 2(\tan(x) \cdot \csc(x) - \cot(x) \cdot \sec(x)) \)

Step 6: Use trigonometric identities to simplify further:

\( \sin(x) + \tan(x) - 1 = 2(\frac{1}{\sin(x)} - \cos(x)) \)

Step 7: Multiply both sides by \( \sin(x) \) to clear the fractions:

\( \sin(x) \cdot \sin(x) + \sin(x) \cdot \tan(x) - \sin(x) = 2 - 2 \cdot \cos(x) \cdot \sin(x) \)

Step 8: Simplify and collect like terms:

\( \sin^2(x) + \sin(x) \cdot \tan(x) - \sin(x) = 2 - 2 \cdot \cos(x) \cdot \sin(x) \)

Step 9: Use the trigonometric identity \( \sin^2(x) = 1 - \cos^2(x) \) to replace \( \sin^2(x) \):

\( 1 - \cos^2(x) + \sin(x) \cdot \tan(x) - \sin(x) = 2 - 2 \cdot \cos(x) \cdot \sin(x) \)

Step 10: Move all terms to one side of the equation:

\( \cos^2(x) + \sin(x) \cdot \tan(x) - \sin(x) - 2 \cdot \cos(x) \cdot \sin(x) + 1 - 2 = 0 \)

Step 11: Combine like terms:

\( \cos^2(x) + \sin(x) \cdot \tan(x) - \sin(x) - 2 \cdot \cos(x) \cdot \sin(x) - 1 = 0 \)

Now, you have a quadratic equation in terms of trigonometric functions. This equation can be further simplified or solved depending on the context of your problem.