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Question: Prove that: \(\operatorname{sec} (\pi - \theta ) = - \sec \theta \) ....

Prove that: sec⁡(π−θ)=−sec⁡θ\operatorname{sec} (\pi - \theta ) = - \sec \theta .

Explanation

Solution

By using the basic trigonometric identity given below we can simplify the above expression that is sec⁡(π−θ)=−sec⁡θ\operatorname{sec} (\pi - \theta ) = - \sec \theta . In order to solve and simplify the given expression we have to use the identities and express our given expression in the simplest form and thereby solve it. Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.

Complete step-by-step solution:
We have to prove that ,
sec⁡(π−θ)=−sec⁡θ\operatorname{sec} (\pi - \theta ) = - \sec \theta ,
To prove: sec⁡(π−θ)=−sec⁡θ\operatorname{sec} (\pi - \theta ) = - \sec \theta
Proof: taking left hand side of the equation,
As (π−θ)(\pi - \theta ) lies in the second quadrant and we know that secant is always negative in the second quadrant , therefore,
⇒sec⁡(π−θ)=−sec⁡θ\Rightarrow \sec (\pi - \theta ) = - \sec \theta
Which is equal to R.H.S ,
Since L.H.S=R.H.S
Hence proved.

Note: Some other equations needed for solving these types of problems , you must know (π−θ)(\pi - \theta ) lies in the second quadrant and we know that secant is always negative in the second quadrant. Range of cosine and sine: [−1,1]\left[ { - 1,1} \right] ,
Also, while approaching a trigonometric problem one should keep in mind that one should work with one side at a time and manipulate it to the other side. The most straightforward way to do this is to simplify one side to the other directly, but we can also transform both sides to a common expression if we see no direct way to connect the two. Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.