MediumMathematicsHigh School (9–10)GeneralTrigonometry › Trigonometric Identitiesproof

Prove that (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A.

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Prove that $(\sin A + \text{cosec}\,A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$ . Then check both sides at $A = 45^\circ$ .

Need a reminder? The key ideaShow

To prove an identity, start with one side and change it step by step into the other. Write ratios in $\sin A$ and $\cos A$ , use $\sin^2 A + \cos^2 A = 1$ , $1 + \tan^2 A = \sec^2 A$ and $1 + \cot^2 A = \text{cosec}^2 A$ , and factorise. One angle where a claim fails shows it is not an identity.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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← For an acute angle A, sec A + tan A = 3.Prove that (1 + tan² A)/(1 + cot² A) = tan² A, and find the value of both sides at A = 60°. →

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