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Trigonometry Practice Test 1

Question : 14 of 40

Marks: +1, -0

\frac{\left(1-\sin A\cos A\right)\left(\sin^2 A-\cos^2 A\right)}{\cos A\left(\sec A-\csc A\right)\left(\sin^3 A+\cos^3 A\right)}

Solution: 👈: Video Solution

Concept:

Simplify using algebraic factorisation and trigonometric identities.

Explanation:

Factorise

\sin^2 A-\cos^2 A

(\sin A-\cos A)(\sin A+\cos A)

Factorise

\sin^3 A+\cos^3 A

(\sin A+\cos A)(1-\sin A\cos A)

Also,

\sec A-\csc A=\frac{1}{\cos A}-\frac{1}{\sin A}=\frac{\sin A-\cos A}{\sin A\cos A}

Substitute these into the expression:

\frac{(1-\sin A\cos A)(\sin A-\cos A)(\sin A+\cos A)}{\cos A\cdot \frac{\sin A-\cos A}{\sin A\cos A}\cdot (\sin A+\cos A)(1-\sin A\cos A)}

Cancel the common factors

(1-\sin A\cos A)

(\sin A-\cos A)

, and

(\sin A+\cos A)

The expression reduces to

\frac{1}{1/\sin A}=\sin A

Answer:

\sin A

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