Studdy AI Logo
Math  /  Trigonometry

Question5) cos⁡θsec⁡θ−sin⁡θcot⁡θ=cos⁡θcot⁡θ−tan⁡θcsc⁡θ\frac{\cos \theta}{\sec \theta}-\frac{\sin \theta}{\cot \theta}=\frac{\cos \theta \cot \theta-\tan \theta}{\csc \theta}

Studdy Solution

STEP 1

1. We are given a trigonometric identity to verify.
2. We need to simplify both sides of the equation to show they are equal.
3. We will use trigonometric identities to simplify expressions.

STEP 2

1. Simplify the left-hand side (LHS) of the equation.
2. Simplify the right-hand side (RHS) of the equation.
3. Compare the simplified LHS and RHS to verify the identity.

STEP 3

Simplify the left-hand side (LHS) of the equation:
The LHS is:
cos⁡θsec⁡θ−sin⁡θcot⁡θ\frac{\cos \theta}{\sec \theta} - \frac{\sin \theta}{\cot \theta}
Recall the identities: sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}.
Substitute these into the LHS:
cos⁡θ1cos⁡θ−sin⁡θcos⁡θsin⁡θ\frac{\cos \theta}{\frac{1}{\cos \theta}} - \frac{\sin \theta}{\frac{\cos \theta}{\sin \theta}}
Simplify each term:
cos⁡2θ−sin⁡2θ\cos^2 \theta - \sin^2 \theta

STEP 4

Simplify the right-hand side (RHS) of the equation:
The RHS is:
cos⁡θcot⁡θ−tan⁡θcsc⁡θ\frac{\cos \theta \cot \theta - \tan \theta}{\csc \theta}
Recall the identities: cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}, tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}, and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.
Substitute these into the RHS:
cos⁡θ⋅cos⁡θsin⁡θ−sin⁡θcos⁡θ1sin⁡θ\frac{\cos \theta \cdot \frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta}}{\frac{1}{\sin \theta}}
Simplify each term:
cos⁡2θsin⁡θ−sin⁡θcos⁡θ1sin⁡θ\frac{\frac{\cos^2 \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta}}{\frac{1}{\sin \theta}}
Multiply by the reciprocal:
cos⁡2θ−sin⁡2θ\cos^2 \theta - \sin^2 \theta

STEP 5

Compare the simplified LHS and RHS:
Both sides simplify to:
cos⁡2θ−sin⁡2θ\cos^2 \theta - \sin^2 \theta
Since the simplified LHS and RHS are equal, the identity is verified.
The identity is verified as true.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord