Holooly Plus Logo
Holooly Plus Logo

Question 6.6.5.8: Establishing an Identity Prove the identity: tan (θ + π/2) =......

Establishing an Identity

Prove the identity: tan⁡ ⁣(θ+π2)=−cot⁡θ\tan\!\left(\theta+{\frac{\pi}{2}}\right)=-\cot\theta

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.
Report Answer

Formula (6) cannot be used because tan⁡π2\tan \frac{\pi}{2} is not defined. Instead, proceed as follows:

tan⁡(α+β) =tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan\left(\alpha+\beta\right)\,={\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}}        (6)

tan⁡(θ+π2)=sin⁡(θ+π2)cos⁡(θ+π2)=sin⁡θcos⁡π2+cos⁡θsin⁡π2cos⁡θcos⁡π2−sin⁡θsin⁡π2\tan\left(\theta+{\frac{\pi}{2}}\right)={\frac{\sin\left(\theta+{\frac{\pi}{2}}\right)}{\cos\left(\theta+{\frac{\pi}{2}}\right)}}={\frac{\sin\theta\cos{\frac{\pi}{2}}+\cos\theta\sin{\frac{\pi}{2}}}{\cos\theta\cos{\frac{\pi}{2}}-\sin\theta\sin{\frac{\pi}{2}}}}

 

=(sin⁡θ)(0) + (cos⁡θ)(1)(cos⁡θ)(0) − (sin⁡θ)(1)=cos⁡θ−sin⁡θ=−cot⁡θ={\frac{(\sin\theta)\left(0\right)\,+\,(\cos\theta)\left(1\right)}{(\cos\theta)\left(0\right)\,-\,\left(\sin\theta\right)\left(1\right)}}={\frac{\cos\theta}{-\sin\theta}}=-\cot \theta

Related Answered Questions

Question: 6.6.5.12

Solving a trigonometric equation linear in sin θ and cos θ Solve asin θ + bcos θ = c (9) ...

Verified Answer:

Divide each side of equation (9) by {\sqrt{...
Question: 6.6.5.11

Solving a trigonometric equation linear in Sine and Cosine Solve the equation: ...

Verified Answer:

Solution A Attempts to use available identities do...
Question: 6.6.5.10

Writing a Trigonometric expression as an algebraic expression Write ...

Verified Answer:

First, for sin⁡−1u\sin^{-1} u, the restric...
Question: 6.6.5.9

finding the exact Value of an expression involving inverse trigonometric functions Find ...

Verified Answer:

We seek the sine of the sum of two angles, ...
Question: 6.6.5.7

Establishing an Identity Prove the identity: tan (θ + π) = tan θ ...

Verified Answer:

\tan\left(\theta+\pi\right)\,=\,{\frac{\tan...
Question: 6.6.5.6

Establishing an Identity establishing the identity: cos(α-β)/sinα sinβ = cot α cot β + 1 ...

Verified Answer:

{\frac{\cos\left(\alpha-\beta\right)}{\sin\...
Question: 6.6.5.5

finding exact Values If it is known that sinα=4/5, π/2< α<π , and that sinβ=-2/√5=-2√5/5, ...

Verified Answer:

(a) Because \sin\alpha={\frac{4}{5}}={\frac...
Question: 6.6.4.8

Establishing an identity Establish the identity: 1-sin θ/cos θ = cos θ/1 + sin θ ...

Verified Answer:

Start with the left side and multiply the numerato...
Question: 6.6.4.7

Establishing an identity Establish the identity: tan v + cot v/sec v csc v =1 ...

Verified Answer:

{\frac{\tan v+\cot v}{\sec v\csc v}} \under...
Question: 6.6.4.6

Establishing an identity Establish the identity: sinθ/1 + cosθ + 1 + cos θ/sin θ = 2csc θ ...

Verified Answer:

The left side is more complicated. Start with it a...