Show that cot θ + cosec θ / sin θ + tan θ = cosec θ cot θ

Question

Asked on March 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-26T14:17:11.332Z

Step 1: Rewrite the left-hand side (LHS) of the equation in terms of and . Recall the identities: = ( )/( ) = (1)/( ) = ( )/( ) Substitute these into the LHS: LHS = ( + )/( + ) = ( )/( ) + (1)/( ) + ( )/( ) Step 2: Simplify the numerator and the denominator separately. Numerator: ( )/( ) + (1)/( ) = ( + 1)/( ) Denominator: + ( )/( ) = ( )/( ) + ( )/( ) = ( + )/( ) Factor out from the denominator: ( ( + 1))/( ) Step 3: Substitute the simplified numerator and denominator back into the LHS expression. LHS = ( + 1)/( )( ( + 1))/( ) Step 4: Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator. LHS = ( + 1)/( ) × ( )/( ( + 1)) Step 5: Cancel out the common term ( + 1) from the numerator and denominator. LHS = (1)/( ) × ( )/( ) Step 6: Multiply the remaining terms. LHS = ( )/(^2 ) Step 7: Rewrite the expression in terms of and . LHS = (1)/( ) × ( )/( ) Using the identities (1)/( ) = and ( )/( ) = : LHS = This matches the right-hand side (RHS) of the original equation. Therefore, the identity is proven. ( + )/( + ) =

Accepted Answer · 2026-03-26T14:17:11.332Z

Step 1: Rewrite the left-hand side (LHS) of the equation in terms of and . Recall the identities: = ( )/( ) = (1)/( ) = ( )/( ) Substitute these into the LHS: LHS = ( + )/( + ) = ( )/( ) + (1)/( ) + ( )/( ) Step 2: Simplify the numerator and the denominator separately. Numerator: ( )/( ) + (1)/( ) = ( + 1)/( ) Denominator: + ( )/( ) = ( )/( ) + ( )/( ) = ( + )/( ) Factor out from the denominator: ( ( + 1))/( ) Step 3: Substitute the simplified numerator and denominator back into the LHS expression. LHS = ( + 1)/( )( ( + 1))/( ) Step 4: Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator. LHS = ( + 1)/( ) × ( )/( ( + 1)) Step 5: Cancel out the common term ( + 1) from the numerator and denominator. LHS = (1)/( ) × ( )/( ) Step 6: Multiply the remaining terms. LHS = ( )/(^2 ) Step 7: Rewrite the expression in terms of and . LHS = (1)/( ) × ( )/( ) Using the identities (1)/( ) = and ( )/( ) = : LHS = This matches the right-hand side (RHS) of the original equation. Therefore, the identity is proven. ( + )/( + ) =

Show that cot θ + cosec θ / sin θ + tan θ = cosec θ cot θ

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Show that cot θ + cosec θ / sin θ + tan θ = cosec θ cot θ

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