Trigonometric Identities
\(\DeclareMathOperator{cosec}{cosec}\)
Identities
-
\[\tan^2\theta+1\equiv \frac{1}{\cos^2\theta}\]
-
\[1+\cot^2\theta\equiv \cosec^2\theta\]
-
\[\tan^2\theta=sec^2\theta-1\]
-
\[\cot^2\theta\equiv \cosec^2\theta-1\]
Solve \(\frac{\tan^2x}{\sec x}=\sec x-\cos x\)
- \(LHS=\frac{sec^2x-1}{\sec x}\)
- \(\frac{\sec^2x}{\sec x}-\frac{1}{\sec x}\)
- \(\sec x-\cos x\)
- \(=RHS\)
Prove the identity \(\frac{\tan^2x}{\sec x+1}\equiv \sec x-\cos^2x-\sin^2x\)
- \(LHS=\frac{\sec^2x-1}{\sec x+1}\)
- \(\frac{\cancel{ (\sec x+1) }(\sec x-1)}{\cancel{ \sec x+1 }}\)
- \(\sec x-1\)
- \(\sec x-(\sin^2x+\cos^2x)\)
- \(\sec x-\sin^2x-\cos^2x\)
- \(=RHS\)
Prove the identity \(\cosec^4\theta-\cot^4\theta\equiv\frac{1+\cos^2\theta}{1-\cos^2\theta}\)
- \(LHS=\cosec^4\theta-\cot^4\theta\)
- \(=(\cosec^2\theta+\cot^2\theta)(\cosec^2\theta-\cot^2\theta)\)
- \(=\cosec^2\theta+\cot^2\theta\)
- \(=\frac{1}{\sin^2\theta}+\frac{\cos^2\theta}{\sin^2\theta}\)
- \(=\frac{1}{\sin^2\theta}+\frac{\cos^2\theta}{\sin^2\theta}\)
- \(=\frac{1+\cos^2\theta}{\sin^2\theta}\)
- \(=\frac{1+\cos^2\theta}{1-\cos^2\theta}=RHS\)