Consider the expression E= , + , +

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Consider the expression

E=\dfrac{\sin\theta\,\sec\theta+\cos\theta\,\csc\theta}{\sec\theta+\csc\theta},

where

\theta

is any angle for which the expression is defined. Show that

E

can be reduced to an unexpectedly simple form. (i) Rewrite each trigonometric ratio appearing in

E

exclusively in terms of

\sin\theta

and

\cos\theta

. Do not perform any other simplification at this stage. (ii) Carry out the algebraic simplifications step by step to obtain a single expression for

E

that no longer contains any reciprocal, quotient, or product of trigonometric ratios. (iii) What is the final simplified form of

E

? State the result clearly and specify the set of values of

\theta

(if any) for which this form is not valid.