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Math  /  Trigonometry

QuestionVerify that the equation is an identity. Show that csc⁡x⋅tan⁡xcos⁡x=sec⁡2x\csc x \cdot \frac{\tan x}{\cos x}=\sec ^{2} x

Studdy Solution

STEP 1

1. The equation involves trigonometric identities.
2. We will use fundamental trigonometric identities to verify the given equation.

STEP 2

1. Express all trigonometric functions in terms of sine and cosine.
2. Simplify the left-hand side of the equation.
3. Compare the simplified left-hand side with the right-hand side.

STEP 3

Express all trigonometric functions in terms of sine and cosine. Start with the left-hand side:
csc⁡x=1sin⁡x,tan⁡x=sin⁡xcos⁡x,andsec⁡x=1cos⁡x\csc x = \frac{1}{\sin x}, \quad \tan x = \frac{\sin x}{\cos x}, \quad \text{and} \quad \sec x = \frac{1}{\cos x}

STEP 4

Substitute these expressions into the left-hand side of the equation:
csc⁡x⋅tan⁡xcos⁡x=(1sin⁡x)⋅(sin⁡xcos⁡xcos⁡x)\csc x \cdot \frac{\tan x}{\cos x} = \left(\frac{1}{\sin x}\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{\cos x}\right)
Simplify the expression:
=1sin⁡x⋅sin⁡xcos⁡2x= \frac{1}{\sin x} \cdot \frac{\sin x}{\cos^2 x}
Cancel sin⁡x\sin x in the numerator and denominator:
=1cos⁡2x= \frac{1}{\cos^2 x}

STEP 5

Recognize that 1cos⁡2x\frac{1}{\cos^2 x} is equivalent to sec⁡2x\sec^2 x:
sec⁡2x=1cos⁡2x\sec^2 x = \frac{1}{\cos^2 x}
Thus, the left-hand side simplifies to the right-hand side:
sec⁡2x=sec⁡2x\sec^2 x = \sec^2 x
The equation is verified as an identity.

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