Question

Evaluate: \[\dfrac{{\sec {{29}^\circ }}}{{\cos ec{{61}^\circ }}} + 2\cot {8^\circ }\cot {17^\circ }\cot {45^\circ }\cot {73^\circ }\cot {82^\circ } - 3\left( {{{\sin }^2}{{38}^\circ } + {{\sin }^2}{{52}^\circ }} \right)\].

Answer 1

Hint: We know that an equation involving one or more trigonometric ratios of unknown angles is called a trigonometric equation. To evaluate the given trigonometric function, as the equation consists of cot functions, as we know that \[\cot \left( {{{90}^\circ } - \theta } \right) = \tan \theta \] hence, applying this function we can split the terms and evaluate it.

Complete step-by-step solution:
 The given function is
 \[\dfrac{{\sec {{29}^\circ }}}{{\cos ec{{61}^\circ }}} + 2\cot {8^\circ }\cot {17^\circ }\cot {45^\circ }\cot {73^\circ }\cot {82^\circ } - 3\left( {{{\sin }^2}{{38}^\circ } + {{\sin }^2}{{52}^\circ }} \right)\]
The equation can be written as
\[\dfrac{{\cos ec\left( {{{90}^\circ } - {{61}^\circ }} \right)}}{{\cos ec{{61}^\circ }}} + 2\cot {8^\circ }\cot {17^\circ }\left( 1 \right)\cot \left( {{{90}^\circ } - {{17}^\circ }} \right)\cot \left( {{{90}^\circ } - {8^\circ }} \right) - 3\left( {{{\sin }^2}{{38}^\circ } + {{\cos }^2}{{38}^\circ }} \right)\]
Simplifying the functions, we get
= \[\dfrac{{\cos ec{{61}^\circ }}}{{\cos ec{{61}^\circ }}} + 2\cot {8^\circ }\cot {17^\circ }\tan {17^\circ }\tan {8^\circ } - 3\left( 1 \right)\]
= \[1 + 2\left( 1 \right) - 3\]
= \[3 - 3\]
= \[0\]

Additional information: In trigonometry sin, cos and tan values are the primary functions we consider while solving trigonometric problems. These trigonometry values are used to measure the angles and sides of a right-angle triangle. Apart from sine, cosine and tangent values, other values are cotangent, secant and cosecant.
Trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. All the trigonometrical concepts are based on these functions. Hence, to understand trigonometry further we need to learn these functions and their respective formulas at first.
If θ is the angle in a right-angled triangle, then
Sin θ = \[\dfrac{{perpendicular}}{{hypotenuse}}\]
Cos θ = \[\dfrac{{base}}{{hypotenuse}}\]
Tan θ = \[\dfrac{{perpendicular}}{{base}}\]

Note: The key point to evaluate any trigonometric function is that we must know all the basic trigonometric functions and their relation. As in the given equation consists of cot functions, hence we must know all the trigonometric identities with respect to the function we need to evaluate. Hence, here are some of the relations:
Tan θ = \[\dfrac{{\sin \theta }}{{\cos \theta }}\], Cot θ = \[\dfrac{{\cos \theta }}{{\sin \theta }}\], Sec θ = \[\dfrac{{\tan \theta }}{{\sin \theta }}\], Cosec θ = \[\dfrac{{\sec \theta }}{{\tan \theta }}\]